Timani Certification Course Journal & Notes Year 1

First day of School!!

Musician's Health & Movement Institute

11 Sep 26

Standard Anatomical Position - standing upright, legs together, arms supinated

All terms, planes, and movements refer back to this position

Anatomical Directions & Terms

    anterior or ventral - from, in, or toward the front

    posterior or dorsal/dorsum/dorsi - from, in, or toward the back

    superior - above or upwards

    inferior - below or downwards

    medial - toward the midline

    lateral - toward the side

    proximal - closer to the body, proximal interphalangeal joint

    distal  - away from the body, distal interphalangeal joint

    superficial - near the surface of the body

    deep - further into the body

Hands & Feet

    plantar - sole of the foot

    palmar - palm of the hand

    dorsum - often used to describe the back of the hand or foot

    ankle supination requires ankle inversion, plantarflexion, and adduction (see below)

    ankle pronation requires ankle eversion, dorsiflexion, and abduction (see below)

Planes of the Body

    sagittal plane - bisects the body symmetrically, divides the body into left and right halves

    sagittal movements travel along this plane (curl into a ball)

        trunk flexion - bow to your audience

        trunk extension - back bend

        elbow flexion - bicep curl

        elbow extension - straightening the arm

        knee flexion - hamstring curl

        knee extension - straightening the leg

        palmar flexion or flexion of the wrist - toward the palm (toward the keyboard)

        dorsiflexion or extension of the wrist - toward the dorsum (away from the keyboard)

        plantar flexion or flexion of the ankle - toward the plantar (toward the pedal)

        dorsiflexion or extension of the ankle - toward the dorsum (away from the pedal)

    coronal plane - divides the body into anterior and posterior halves

    coronal movements travel along this plane (snow angels)
        abduction of the leg or arm - moving limbs away from the body

        adduction of the leg or arm - moving limbs toward the body

        these are called "a-b-duction and adduction" because they sound so similar

        lateral flexion of the spine - reaching sideways to the ground

        ankle inversion - plantars face one another

        ankle eversion - plantars face away from one another

    transverse plane - divides the body into superior and inferior halves

    transverse movements are the only movements that twist or rotate (owl head)

        rotation of the spine

        rotation of the hip joint

        rotation of the knee joint (even though it's mostly a hinge joint in sagittal)

        wrist supination - ulna and radius parallel

        wrist pronation - ulna and radius cross

Scapulae Motions

    scapulae - floating bones along the back that connect the clavicle and humerus

    they're operated by 25 muscles

        elevation - upwards

        depression - downwards

        retraction - medial, toward one another

        protraction - lateral, away from one another

        upward rotation - as the arm abducts in the coronal plane

        downward rotation - as the arm adducts in the coronal plane

Opposition

    thumbs reaching across the hand to touch other fingertips, especially 5

12 Sep 26

It's practical to study a standard body

In reality, our bodies are all different and also shaped by how we've used them

They're not uniform; for example, some people have more vertebrae or ribs

Skeleton

    cranium

    clavicula, scapula, sternum, costae

    humerus, ulna, radius, carpals, metacarpals, phalanges (proximal, middle, distal)

    spine (cervical, thoracic, lumbar), sacrum, coccyx or os coccygis

    ilium

    femur, patella, tibia, fibula, tarsals, metatarsals, phalanges (proximal, middle, distal)

    the heel is a huge tarsal called the calcaneus

    all phalanges are labeled like piano fingering

Spine

    cervical spine - neck, C1-C7

        C7 often sticks out a bit more

    thoracic spine - chest, T1-T12

        T1-T7 are attached to 7 pairs of true ribs, which are attached to the sternum

        T8-T10 are attached to 3 pairs of false ribs

        T11-T12 are attached to 2 pairs of floating ribs

    lumbar spine - lower back, L1-L5

    vertebrae are cushioned with intervertebral discs

    these discs are made of tough tissue with crisscross fibers built for shock absorption

    the sacrum is part of the spine, 5 fused bones

    it has holes for nerves to pass through

    the coccyx is part of the spine, 3-5 semi-fused bones

    the coccyx attaches to all the pelvic floor muscles

    with excessive tension, our tight pelvic floors pull our tails between our legs

Ball & Socket

    before we walked on two feet, we were on all fours

    because of this, there are many structural similarities between the arms and legs

    hips and shoulders are ball and socket joints

    hip - ball is the head of the femur; socket is the acetabulum of the pelvis

    shoulder - ball is the head of the humerus; socket is the glenoid cavity of the scapula

Synovial Joints

    the ends of these bones are covered in articular cartilage, which is smooth and slippery

    it absorbs shock and reduces friction so bones can move smoothly against each other

    between the bones is a small space called the joint cavity

    this space holds synovial fluid, which lubricates the joint and reduces friction

    synovial fluid also nourishes the articular cartilage

    the joint capsule is a tough, flexible sleeve that surrounds the synovial joint

    the joint capsule has two layers:

        the outer layer provides stability

        the inner synovial membrane produces synovial fluid

    out of the joint capsules grows ligaments

Types of Synovial Joints

Ligaments

    ligaments are very strong bands of connective tissue that connect bone to bone

    they stabilize joints, prevent excessive movement, and keep bones properly aligned

    they're less elastic than tendons

    if a bone goes out of joint, it is likely that the ligament has torn

Misc.

    intercostals makes sense when you know the costa

    iliac crest makes sense when you know the ilium

    ulnar deviation makes sense when you know the ulna

    ulnar deviation is possible because the ulna and the carpals aren't connected

    the wrist joint is between the radius and the carpals

    the shoulder girdle is the clavicle and scapula

    the patella floats inside a tendon

Introduction to Muscles

    we have over 600 muscles, but we won't learn about all of them

    muscles can only contract and relax, like a switch

    a muscle can NOT cause both flexion and extension because it cannot actively lengthen

    it will always try to do all of its functions

    if only one function occurs, it's because other muscles are stabilizing

    chocolate muscles are not bad muscles; we need them for mobilizing

    they are mobilizers instead of stabilizers

    3 types of contraction - concentric, eccentric, isometric

    origin of a muscle is closer to the midline or higher in the body

    insertion of a muscle is further from the midline or lower in the body

Abbreviations

    simplified origin - SO

    simplified insertion - SI

    primary function - PF

    secondary function - SF

    Timani notes - TMN

Pectoralis Major

    SO: clavicle, sternum, abdominal fascia

    SI: upper humerus

    PF: adduction of the shoulder joint

    PF: horizontal adduction of the shoulder joint (starting from lifted arm)

    SF: medial rotation of the shoulder joint

    TMN: chocolate muscle

        relax as much as possibly to release sound and free the arm

Deltoideus

    3 parts - anterior, mid, posterior

    SO: clavicle, acromion, scapular spine

    SI: upper humerus

    anterior/clavicular

        PF: flexion of the shoulder joint

        SF: medial rotation of the shoulder joint

    posterior/spinal

        PF: extension of the shoulder joint

        SF: lateral rotation of the shoulder joint

    mid/acromial

        PF: abduction of the shoulder joint

    TMN: chocolate muscle

        needed for specific movements, can be offloaded with push-off

13 Sep 26

Biceps Brachii

    "2 headed of the arm" (triceps are 3 headed, quadriceps are 4 headed)

    SO: tubercle & coracoid process of the scapula

    SI: upper radius & fascia

    PF: flexion of the elbow joint

    PF: supination

    SF: flexion of the shoulder joint

    TMN: chocolate muscle

        can be offloaded with the back arm-line

Brachialis

    SO: mid humerus

    SI: upper ulnar

    PF: flexion of the elbow joint

    TMN: caramel muscle

        brachialis & supinator are deeper than biceps

Quadriceps Femoris

    rectus femoris (middle, largest muscle, covers most of the other 3)

    vastus lateralis (lateral side of femur)

    vastus medialis (medial side of the femur)

    vastus intermedius (in front of the femur, between the other two)

    the quadriceps, like the biceps, cross 2 joints

    SO: rectus femoris - ilium; all 3 vasti - upper femur

    SI: - through patellar ligament to upper tibia

    PF: extension of the knee joint

    PF: rectus femoris only - flexion of the hip joint

    TMN: chocolate muscle

        except for vastus medialis, the only quad muscle in caramel

Tibialis Anterior

    SO: upper tibia

    SI: medial cuneiform

    PF: dorsiflexion

    SF: inversion of the ankle joint

    TMN: chocolate muscle

        often undifferentiated from the quads

Fibularis Longus

    SO: upper fibula

    SI: medial cuneiform

    PF: plantar flexion

    SF: eversion of the ankle joint

    TMN: chocolate and caramel muscles

        chocolate for eversion, caramel for supporting transverse arch

Tensor Fascia Latae

    SO: iliac crest

    SI: iliotibial (IT) band

    PF: abduction of the hip joint

    PF: medial rotation of the hip joint

    TMN: chocolate muscle

        leg rotator exercise, frees hip

Adductors

    this is a group of 6 main muscles:

        adductor longus

        adductor brevis

        adductor magnus

        pectineus

        gracilis

        obturator externus

    SO: pubis

    SI: femur; gracilis attaches to upper tibia

    PF: adduction of the hip joint

        some adductors - flexion of the hip joint

        some adductors - extension of the hip joint

        some adductors - medial rotation of the hip joint

    TMN: chocolate muscle

        squeezer exercise, frees breathing with lower chest expansion

Obliquus Externus    

    SO: outer surface of ribs 5-12

    SI: outer lip of iliac crest, linea alba

    PF: rotation of the spine

    PF: lateral flexion of the spine

    TMN: chocolate muscle

        can overengage in exhalation

        good muscle for lateral arm movements

        twisting to the right activates the left external oblique and the right internal oblique

        external obliques run in the direction of hands in pockets in front of you

        internal obliques run in the direction of hands in pockets behind you

Rectus Abdominis

    SO: cartilage of ribs 5-7, xiphoid process of the sternum

    SI: pubis

    PF: flexion of the spine

    TMN: chocolate muscle

        can restrict ribs/breathing

Serratus Anterior

    SO: ribs 1-9

    SI: scapula

    PF: protraction of the scapula

    PF: stabilization of the scapula

    TMN: caramel muscle

        stabilizes scapula/shoulder

18 Sep 26

Trapezius

    3 parts - upper, middle, lower

    upper/descending

        SO: occipital bone, nuchal ligament

        SI: clavicle

        PF: elevation of scapula to neutral

        PF: stabilization of shoulder girdle

        PF: stabilization of the neck

    middle/transverse

        SO: broad aponeurosis at T1-T4

        SI: acromion

        PF: retraction of the scapula

        PF: stabilization of the scapula during many arm movements

    lower/ascending

        SO: spinous processes of T5-T12

        SI: scapular spine

        PF: upward rotation of the scapula

        PF: then depress scapula when upwardly rotated

    TMN: as a whole, the muscle stabilizes scapulae against the thoracic cage

        important to stabilize neck and shoulder girdle,

        provide stability behind the movements of the arm,

        keeps head from moving forwards

        

Latissimus Dorsi

    SO: spinous processes of T7-T12, iliac crest, ribs 9-12, scapulae

    SI: upper humerus

    PF: adduction of the shoulder joint

    PF: extension of the shoulder joint from a flexed position

    SF: medial rotation of the shoulder joint

    TMN: caramel muscle

        activation of the "back arm pit" anchors the arm/shoulder

        helps playing chords at piano and bowing for violin and viola

        "playing from the back," part of the back arm-line push-off

        overuse can restrict shoulder flexibility

Teres Major

    SO: scapula

    SI: upper humerus

    PF: medial rotation of the shoulder joint

    PF: adduction of the shoulder joint

    TMN: chocolate muscle

        overuse can keep arm clenched into the side

        lats do not do this clenching

Rhomboideus

    2 parts - major and minor, but they do the same thing; minor are superior

    SO: rhomboid minor C6-C7; rhomboid major T1-T4

    SI: medial border of the scapula

    PF: retraction of the scapula

    PF: downward rotation of the scapula

    TMN: chocolate muscle

        often overused to try to create a better posture; over rotates scapulae

        serratus anterior is a better stabilizer

19 Sep 26

SITS muscles

    4 parts - supraspinatus, infraspinatus, teres minor, subscapularis

    supraspinatus

        SO: supraspinous fossa of the scapula

        SI: greater tubercle of the humerus

        PF: abduction of the shoulder joint

    infraspinatus

        SO: infraspinous fossa of the scapula

        SI: greater tubercle of the humerus

        PF: lateral rotation of the shoulder joint

    teres minor

        SO: lateral border of the scapula

        SI: greater tubercle of the humerus

        PF: lateral rotation of the shoulder joint

        SF: adduction of the shoulder joint

    subscapularis

        SO: subscapular fossa of the scapula

        SI: lesser tubercle of the humerus

        PF: medial rotation of the shoulder joint

    TMN: caramel muscle

        the infraspinatus is often too tense, weakening/stretching the subscapularis

        all the muscles of the rotator cuff stabilize the shoulder joint

        they also keep the humeral head in the joint socket

Triceps Brachii

    SO: scapula, upper humerus, lower humerus

    SI: upper ulna

    PF: extension of the elbow joint

    PF: extension of the shoulder joint

    SF: adduction of the shoulder joint

    TMN: deep part is more caramel, other parts are chocolate

        important part of the back arm-line

        connects ulnar side of hand and wrist through the upper arm and shoulder blade

Extensor Carpi Ulnaris

    SO: ulna, lower humerus

    SI: ulnar side of 5th metacarpal

    PF: dorsiflexion on the ulnar side

    PF: radial deviation (adduction of the wrist)

    TMN: caramel muscle

        can control wrist movement without becoming stiff

        balances with stronger thenar side of the hand

        (there are two more extensors of the wrist: carpi radialis longus & brevis)

Gluteus Medius

    SO: below iliac crest

    SI: upper femur

    PF: abduction of the hip joint

    PF: stabilizer for standing, walking, & climbing stairs

        anterior part - flexion and medial rotation

        posterior part - extension and lateral rotation

    TMN: caramel muscle, very good muscle to use

        clam exercise, hydrant exercise, standing shuffle

        for musicians, very good for stability in standing

        this muscle relaxes if we hang into the hip forward position

Gluteus Maximus

    SO: sacrum, ilium

    SI: upper fibers - IT band lower fibers - upper femur

    PF: extension of the hip joint

    PF: lateral rotation of the hip joint

    TMN: heel walk

        chocolate muscle - upper fibers are superficial mobilizers - abduction of the hip joint

        caramel muscle - lower fibers are deep stabilizers - adduction of the hip joint

        (there is also a gluteus minimus)

        connection with the floor through the heel to gluteus maximus, helps with feeling grounded

Hamstrings

    3 parts - biceps femoris, semitendinosus, semimembranosus

    biceps femoris

        SO: ischial tuberosity, mid femur

        SI: head of fibula

        PF: flexion of the knee joint

        PF: extension of the hip joint

        SF: adduction of the hip joint

        SF: lateral rotation of the knee joint

    semitendinosus & semimembranosus

        SO: ischial tuberosity

        SI: upper tibia & connecting tissue

        PF: flexion of the knee joint

        SF: medial rotation of the knee joint

    TMN: mostly a chocolate muscle

        when tight, they pull the pelvis into a posterior tilt in both standing and sitting

        can interfere with grounding

Gastrocnemius

    SO: lower femur

    SI: calcaneal tuberosity (Achilles tendon)

    PF: plantar flexion

    PF: flexion of the knee joint

    TMN: chocolate muscle

        can interfere with grounding

        Achilles tendon stores kinetic energy like a spring

Soleus

    SO: upper fibula

    SI: calcaneal tuberosity

    PF: plantar flexion only

    TMN: caramel muscle

        triceps surae: gastrocnemius & soleus; soleus is under gastrocnemius

        helps ground the forefoot, keeps body from falling forwards when standing

        stretch with bent knees, as it doesn't cross the knees like the gastrocnemius

        if under activated, lack of stability in the ankle joint

20 Sep 26

Schlitterbahn Tips

Yesterday, we went to Schlitterbahn for my birthday.  We've been several times before, but so many things went right that I wanted to write them down.

Shoes

I've done this with bare feet, sandals/crocs, aqua socks, and old sneakers with no socks.  The best is any shoe that stays on your foot no matter what, which means sandals with straps around the ankle, aqua socks, or sneakers.  Sandals give you another surface to sunscreen, so the the two options are better.  Aqua socks can get pretty beat up from the wear and tear, so an old pair of sneakers with a netted top is the most practical bet.

Swim Bottoms

My newest pair of swim bottoms have a camel toe seam, so I decided to wear a well-constructed pair of period underwear under them.  The double layer was an accident, and I never want to go back.  Nothing rides up, nothing goes out of place as you get thrown down Whitewater, and you just don't worry about it.  Wedgie free.

Swim Tops

Another accident - one of the rash guards on sale this year was long sleeved and hooded.  I bought it without thinking too much about the hood, but I will from now on.  No hat to lose, and no sunscreen on the neck.  Hair has to be down in pigtails so the hood stays up easily.  Any comfortable bikini top underneath, and you're good to go.

Croakies

Sunglasses are a must, and they also get thrown off your face.  Floating croakies, tightened, are the way to go; my sunglasses never came off, but more importantly, I wasn't worried about it.

Waterproof Phone Carrier

We've known this was a necessity for years.

Cooler

This year, we had some topo chicos, sandwiches, and berries.  It was perfect - not too heavy, not too messy, and not too much.  Sunscreen in the cooler to reapply, and finish your topo chico before heading back out.

Towels

Bring them and leave them in the car.

Timing

This was the first year we went on a Sunday, and after a full week of school had passed.  The hours were shorter, but the crowds were far more manageable.  Worth it.  Arrive 15-25 mins before the park opens.

Fastlane

Yes, but no need to buy the fancy one.  The cheaper 1-pass-per-ride is much better; no need to use it at the beginning or the end of the day.  Never ride Wolfpack; it has no fastlane, and the line doesn't move.  Regarding lines, riddles or situational puzzles make the time go by much faster.

I hope my little leech always wants to ride with her Mama

Elementary Combinatorics

Dad was being annoying again.  He told me not to write a post on permutations, combinations, Pascal's triangle, and binomial coefficients, and then he said, "Just find some time to have a conversation with Drakeson about it."  But in order to have a conversation, I need to know what I'm talking about, and for that to happen, I have to write a post.

Combinatorics is a branch of math concerned with counting and arranging.  Most word problems that begin with "how many ways" are combinatorics problems.  Did you know that there are about 10⁸⁰ atoms in the universe and about 10¹²⁰ different games of chess?  So for every atom in the universe, there are about 10⁴⁰ games of chess.  Welcome to the wild world of combinatorics.

The difference between combinations and permutations is that combinations are situations where order does not matter, and permutations are situations where it does.  If you have a collection of things, and you don't arrange them in any particular way, this is a combination.  Every bag of marbles is a combination.  If there is an arrangement, you have a permutation.  So the next time your child makes you a beaded bracelet, thank them for the permutation.

---

There are four categories of counting and arranging.  They are:

Permutations where repetition is not allowed

Permutations where repetition is allowed

Combinations where repetition is not allowed

Combinations where repetition is allowed

Dad likes to talk about how many different ways there are to arrange people at a dinner table.  That's a permutation where repetition is not allowed.

I like to talk about flavors of ice cream scoops in an ice cream cone.  That's a permutation where repetition is allowed.

Milli tries to bring too many bottles of hand sanitizers on the plane to WA for summer vacation.  She's stopped by TSA and has to make some decisions.  That's a combination where repetition is not allowed.

When I was growing up, Debbie baked a dozen each of 6 types of Christmas cookies.  She would give me a plate and tell me that I could choose any 4.  That's a combination where repetition is allowed.

---

Dad's family was: Bill, Ruth, William, Mary, Tommy, Steve, and Pete.  We'll pretend they're knights and arrange them at a round table, placing them in seats numbered 1, 2, 3, 4, 5, 6, and 7.  Let it be known that seat 1 is the head of the table.  How many different ways can the Kratzkes arrange themselves?

There are 7 ways to choose the person who sits in seat 1.  Because there are only 6 people left, there are now 6 ways to choose the person who sits in the seat 2.  So there are 7*6 different ways to choose Kratzkes for seats 1 and 2.  For each of these 42 options, there are 5 choices for seat 3, which means there are 7*6*5 ways to designate Kratzkes into the first 3 seats.  Continuing this pattern, there are 7 factorial, notated 7!, and meaning 7*6*5*4*3*2*1 ways to seat this family at the round table.

By the by, mathematicians have decided that 0! = 1, which means there is 1 way to order 0 people at a round table.

But what if the round table is pretty small, and it only seats 4 people?  How many ways can we seat 4 members of the family?  (Seat 1 is still the head of the table.)  Well, like before, there are 7 ways to fill seat 1 and 6 ways to fill seat 2.  With 4 seats, we have 7*6*5*4 ways to fill them.  This is the same as 7!/3!.

Permutations with no repetitions:  n! / (n - r)! , where we're choosing r out of n things.

---

I love Baskin Robbins.  Their logo highlights the number 31 because they have 31 flavors.  Let's say we're in the mood for a cone, and we can eat 5 scoops of ice cream.  Order absolutely matters; chocolates, malts, caramels, and vanillas are more enhanced with the taste and texture of a waffle cone than fruitier flavors.  Furthermore, it's kind of annoying to have two brightly flavored scoops touching; nobody would argue with the sensible decision of placing chocolate between mint and strawberry.

We have the choice of 31 flavors for our first scoop.  For each of those choices, we have the choice of 31 flavors our second scoop.  So there are 31*31 or 31² different ways to choose our first two scoops.  Continuing this line of thinking, there are 31⁵ different ways to choose 5 scoops of ice cream.

Multiplying everything together also works when you don't have the same number of things to choose from every time.  Imagine that your significant other finds you on your stroll home.  His pocket doesn't quite conceal the outline of a newly purchased jewelry box, his palms are sweaty, and trying to act nonchalant, he asks you to dinner.  Before you can explain that you just ate no fewer than 5 enormous scoops of triple fudge chocolate ice cream, you find yourself gazing upon a prix fixe menu.  You may choose 1 of 18 wines, 1 of 4 salads, 1 of 3 appetizers, 1 of 1 palette cleansers, 1 of 5 entrées, 1 of 4 personal sides, 1 of 3 cheese plates, and 1 of 4 desserts.  You want the health insurance, so you're going to say yes.  To make him feel like the luckiest guy in the restaurant, you explain that each of you has the choice of 18*4*3*1*5*4*3*4 dinners, scribbling down the beginnings of a tree diagram on his beverage napkin.

---

If combinations are like permutations, but order does NOT matter, then there ought to be fewer combinations than permutations.

Milli has acquired 24 different types of hand sanitizers, all in 2 oz bottles.  When packing for her trip to WA, she has chosen to bring them all.  The transportation security officer, Madge, explains the 3-1-1 liquid rule: bottles must be 3.4 oz or smaller, they must all fit into a 1 quart sized clear plastic bag, and each passenger is allowed exactly 1 bag.  Milli realizes she can fit 6 hand sanitizers into a quart sized bag and asks Madge how many ways she can choose 6 from 24.

Charmed, Madge replies, "First we need to pretend that order matters and calculate the permutation.  Then we'll make that number smaller by dividing in order to get rid of the duplicates.  For our first hand sanitizer, we have 24 choices.  For our second, 23.  So to select all 6 bottles, we have 24*23*22*21*20*19 choices, which can also be expressed as 24!/(24-6)!."

Milli nods in agreement.  Madge continues.  "But because we don't care about order, we have to divide by something to knock out the duplicate sets.  Do you know how many ways there are to order each set of 6 hand sanitizers?"  Milli replies, "Of course.  Six factorial."  Madge nods approvingly and continues, "So if we have 6! duplicates, we need to divide 24!/(24-6)! by 6!, which yields: (24*23*22*21*20*19)/(6*5*4*3*2*1).  And because the answer to "how many ways" is always an integer, these fractions will always simplify elegantly."

Milli thanks Madge, procures 4 clear quart sized bags out of nowhere, packages up 4 bags of 6, and distributes them among her dad, mom, and brother.

Let's return our attention to the Kratzkes at the Restaurant of the Round Tables.  They had such a wonderful time counting permutations last week that they have returned.  But tonight is different - tonight, the hostess has announced that the table for 4 is available, but the table for 7 is occupied for the evening due to some grisly game involving the beheading of potential suitors.  Unperturbed, the Kratzkes notice that there's a brand new carnival & batting cage next door, and to attract future business, it's running a promotion where everybody with the letter "z" in their last name attends free of charge.  This means they're now faced with the decision: which 4 of them will have dinner, and which 3 will go to the carnival?  Oh, how the tables have turned!  What was once a permutation problem has now become a combination problem.

Let's focus on one question - which 4 of them will have dinner?  We remember from before that there were 7!/3! ways to fill the 4 person table, but for the next few minutes, we don't care about who goes in which seat.  For each selection of 4, there are 4! duplicates, so we must divide 7!/3! by 4!.

Combinations with no repetitions:  n! / [r!(n - r)!] , where we're choosing r out of n things.

This situation is called "n choose r."

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Before we get to combinations in which repetitions ARE allowed, let's dwell a moment on the combinations-with-no-repetitions phrase "n choose r."

It's all written out in Pascal's triangle.

These hexagonal Pascal's triangle graphics were taken from Archimedes Lab.

Powers of 2

When we're looking for the number "n choose r," we can find it by going down to row (n + 1) and finding the term (r + 1).  In the latest case of the Kratzkes, (7!)/(4!3!), we can go down to the 8th row (1, 7, 21, 35, 35, 21, 7, 1) and find the 5th term (35).

The pattern that emerges when reading Pascal's Triangle from left to right is also called "the binomial coefficient."

A polynomial is an expression made of variables and coefficients using addition, subtraction, multiplication, and powers to non-negative integers.  Here's an example of a polynomial: 4a³b -  b² + 1.  The only reason people get so excited about polynomials is that when polynomials are added together, they yield more polynomials.

A binomial is a polynomial with two terms.  For example, (a + b) is a really good one.  We can make other ones, like (4c⁵ - 6.789), but let's stick with (a + b).

There's a trick to expanding (a + b)ⁿ.  The expansion yields a string of variables attached to the coefficients from row (n + 1) of Pascal's Triangle.  For example,

(a + b)⁹ =  a⁹ + 9a⁸b + 36a⁷b² + 84a⁶b³ + 126a⁵b⁴  + 126a⁴b⁵  + 84a³b⁶  + 36a²b⁷ + 9ab⁸  + b⁹.

Those coefficients are in row 10.

These strings of numbers in Pascal's triangle, read from left to right, also represent the binomial distribution.  There's a physical board that approximates the binomial distribution, and you've probably seen it.  It's called a quincunx or a Galton board, and they use one under the name "Plinko" in The Price is Right.  These are peg boards in which a ball has an equal chance of bouncing left or right at each junction.

Let's switch our thinking from the quincunx to the classic binomial prop - the penny.  The more pennies we flip, the less likely it is we'll flip all heads or all tails, and the more likely it is that our heads and tails counts will be closer together.  If we look at the 12th row of Pascal's triangle, we can see the probability of getting some number of heads and tails if we toss a penny 11 times.  In 1 out of 2¹¹ times, we'll flip all tails.  In 11 out of 2¹¹ times, we'll flip exactly 10 heads and 1 tail.  In (330 + 462 + 462 + 330) out of 2¹¹ times, we'll flip 4, 5, 6, or 7 heads out of 11 chances.  That's a little over 77% of the time.

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...“and what is the use of a book,” thought Alice “without pictures or conversations?”

Addition

Squares & Sums

Integers - line segments or 1D triangles

Triangular Numbers - triangular grid in 2D (like bowling pins)

Tetrahedral Numbers - pyramids in 3D where all sides are triangular numbers

Pentatope Numbers - hard to understand; triangles in 4D

The Fibonacci Sequence

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Debbie's cookies were a marvel.  I had a particular fondness for the peanut butter blossoms, and Angel loved the chocolate crinkles.  Amongst the others were peppermint candy cookies, green tree almond press cookies, frosted maraschino chocolate cookies, and Aunt Katherine's caramel cups.  You might recall that Debbie said I'm allowed to choose any 4.

Aunt Katherine's - A

Blossoms - B

Chocolate Crinkles - C

Drop Peppermints - D

Evergreens - E

Frosted Maraschinos - F

There's a reason this chapter is last; combinations in which repetitions are allowed are the trickiest of them all.

We can't take 6⁴ and then divide something out, because the cases can't agree on what that number would be.  For example, imagine I chose 4 Aunt Katherine's (A A A A).  There are no duplicates for that pattern.  But what if I chose (B C D E)?  There are 4! duplicates for that one.

Nor can we pretend the set is (A A A A B B B B C C C C D D D D E E E E F F F F) and choose 4.  The permutation would be 24*23*22*21, but just like before, we wouldn't know what to divide by.  Again, (A A A A) would yield no duplicates and (B C D E) would yield 4!.

To figure this one out, we actually have to morph the problem into a different "n choose r" situation.  (Now would be a good time for me to thank the website mathisfun, which has been guiding me through writing this entire post.)

Let's put Debbie's cookies onto her Christmas cookie platters.  Now they look like this:

A

B

C

D

E

F

Instead of tracking our cookies, we're going to write a code that describes our actions.

M - we MOVE to the next plate, taking to cookies.

T - we TAKE 1 cookie and stand still, stupidly staring at the same plate.

So if I choose 4 Aunt Katherine's, that looks like this (T T T T M M M M M).  And if I choose a blossom, a chocolate crinkle, a drop peppermint, and an evergreen, that looks like this (M T M T M T M T M).

This may seem convoluted, but here's where it pays off: both of these strings are 9 digits long.  By tracking our actions instead of our cookies, we can represent each choice in a way that lends itself to factorials and division once again.  The string will always be exactly r + (n - 1) long, where r is the number of cookies Debbie is letting us have, and n is the number of types of cookies she has made.

How many ways can we distribute T's?  Our question has changed topics from a set of 6 cookies to a string of 9 actions, but we're still choosing 4.  And "9 choose 4" on Pascal's triangle is row 10 term 5, which is 126.  Or we can use our "n choose r" formula to get 9!/(4!5!) = 126.

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choosing r out of n	repetition not allowed	repetition allowed
permutations, order matters	n! / (n - r)!	nr
combinations, order doesn't matter	n! / [r!(n - r)!]	(r + n - 1)! / [r!(n - r)!]

After the Decimal

This week, Dad asked me to find an algorithm for turning base 10 decimals into binary.  He's so annoying.

To review what binary is, we first need to review base 10.  We have a counting system called base 10, where we organize our things into piles of 10's using digits 0, 1, 2, 3, 4, 5, 6, 7, 8, & 9.  Once we get too many of those piles, we group 10 of them together and call them "hundreds."  And we organize those into bigger piles of 10 hundreds, and those are called "thousands."  This goes on for quite some time - enough to take care of our counting needs, anyway.  We stop understanding numbers easily by one million, which is why we haven't already started a revolution, despite the fact that one person in our country has a trillion dollars.

10³ - one thousand

10⁶ - one million

10⁹ - one billion

10¹² - one trillion
10¹⁵ - one quadrillion
10¹⁸ - one quintillion
10²¹ - one sextillion

10²⁴ - one septillion
10²⁷ - one octillion
10³⁰ - one nonillion
10³³ - one decillion

But the choice of the number "10" and the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, & 9 aren't based on some sort of universal truth, and in fact, they're a bit random.  Binary or base 2 is what we call it when we start organizing our things into piles of 2 instead.  We only need two digits when we do that, so we use 0 and 1.  We can group those piles of 2 into bigger piles of two 2's, and the unit 2² is like our hundreds unit.  The unit 2³ is like our thousands.

Here's a helpful exercise.  Let's just count a few numbers in binary and see what they look like.

1 - 1

2 - 10

3 - 11

4 - 100

5 - 101

6 - 110

7 - 111

8 - 1,000

9 - 1,001

10 - 1,010

11 - 1,011

12 - 1,100

13 - 1,101

14 - 1,110

15 - 1,111

Let's explore the number: 111,111.

In base 10, that number means 1 pile of 10⁵ + 1 pile of 10⁴ + 1 pile of 10³ + 1 pile of 10² + 1 pile of 10¹ + 1 pile of 10⁰.  In base 2, it means 1 pile of 2⁵ + 1 pile of 2⁴ + 1 pile of 2³ + 1 pile of 2² + 1 pile of 2¹ + 1 pile of 2⁰.  (Notice that our ones units are counting ones in both systems, because 10⁰ = 2⁰ = 1.)

Now consider this number: .111

In base 10, that's one tenth and one one-hundredth and one one-thousandth.

Another way to look at this is one 10^-1 + one 10^-2 + one 10^-3.

In base 2, that's one half and one fourth and one eighth, or one 2^-1 + one 2^-2 + one 2^-3.

So back to Dad's question.  Let's turn some long ugly number with a decimal into binary.

I'll choose 654.32̅1̅.

(If you were dying to convert an irrational number into binary and found yourself reading this post, I'm sorry to tell you that you'll have to round.  Life is full of disappointments, and you may as well get used to it now.)

The first step is to work on the part to the left of the decimal.

We can call that the before-the-decimal algorithm.

Divide 2 into the integer and write the remainder in the ones (2⁰) unit.

654/2 = 327 R 0

0.

That's the whole algorithm, so now we'll repeat it for the next unit to the left.

Divide 2 into the new number (327) and write the remainder into the 2's (2¹) unit.

327/2 = 163 R 1

10.

The next remainder goes into the 4's (2²) unit.

163/2 = 81 R 1

110.

8's (2³) unit.

81/2 = 40 R 1

1,110.

16's (2⁴) unit.

40/2 = 20 R 0

01,110.

32's (2⁵) unit.

20/2 = 10 R 0

001,110.

64's (2⁶) unit.

10/2 = 5 R 0

0,001,110.

128's (2⁷) unit.

5/2 = 2 R 1

10,001,110.

256's (2⁸) unit.

2/2 = 1 R 0

010,001,110.

512's (2⁹) unit.

2/1 = 0 R 1

1,010,001,110.

That takes care of the part before the decimal, but before we go on, we ought to check our work.

512 + 128 + 8 + 4 +  2 = 654

Now we turn our attention to the part after the decimal.

It is: .32̅1̅

Did you know that any repeating decimal can be expressed as a fraction?

Let's do that; I know a trick.

First, name your repeating decimal.  I'll call this one x.

x = .32̅1̅

Then multiply it by 10^whatever so you're set up to subtract and erase the repeating part.

100x = 32.12̅1̅

so

100x - x = 31.8

and

99x = 31.8

which means

x =  318/990 = 106/330 = 53/165

We're all ready to get started!

The after-the-decimal algorithm is quite similar to the before-the-decimal one.

We'll be working from the unit nearest the decimal and then moving to the right.

Multiply the fraction by 2.

If the result is ≥ 1, put a 1 in the halves (2^-1) unit, and if it is not, put in a 0.

Once the fraction becomes greater than 1, subtract 1 to find the new fraction.

We start with the 1/2's or (2^-1) unit with the fraction 53/165.

53*2/165 = 106/165 < 1

1,010,001,110.0

1/4's (2^-2) unit.

106*2/165 = 212/165 ≥ 1

1,010,001,110.01

1/8's (2^-3) unit.

47*2/165 = 94/165 < 1

1,010,001,110.010

1/16's (2^-4) unit.

94*2/165 = 188/165 ≥ 1

1,010,001,110.0101

1/32's (2^-5) unit.

23*2/165 = 46/165 < 1

1,010,001,110.01010

1/64's (2^-6) unit.

46*2/165 = 92/165 < 1

1,010,001,110.010100

1/128's (2^-7) unit.

92*2/165 = 184/165 ≥ 1

1,010,001,110.0101001

1/256's (2^-8) unit.

19*2/165 = 38/165 < 1

1,010,001,110.01010010

1/512's (2^-9) unit.

38*2/165 = 76/165 < 1

1,010,001,110.010100100

1/1024's (2^-10) unit.

76*2/165 = 152/165 < 1

1,010,001,110.0101001000

1/2048's (2^-11) unit.

152*2/165 = 304/165 ≥ 1

1,010,001,110.01010010001

That looks like a pretty good place to stop, so we can truncate and call it a day.

I would have stopped at the 1/512's (2^-9)  unit, but I thought I'd wait for the next 1 to lock in that 2,048th.

As Skeletor says, "Until we meet again!"