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All my math notes, now in Markdown.

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Permutations and Combinations

Learning Targets

You should be able to

Concepts / Definitions

The Fundamental Counting Principle states that if one event has $m$ possible outcomes and a second independent event has $n$ possible outcomes, then there are $m$ x $n$ total possible outcomes for the two events together.

If you have four flavors of ice cream and two types of cones, then ther are $4 * 2 = 8$ possible combinations.

Ice Cream Flavors and Toppings

In mathematics, the factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$. \(n! = n(n-1)(n-2)(n-3)\ ...\ (2)(1)\) By definition, $0! = 1$.

Permutations are the number of ways a set of $n$ distinguishable objects can be arranged in order.
$4!$ = 24 ways to order four items

Permutation of the letters ABCD

The number of permutations on $n$ objects taken $r$ at a time is given by \(P\binom{n}{r} = P(n, r) = nPr = \frac{n!}{(n-r)!}\)

The number of ways $n$ items can be ordered with replacement $r$ times is $n^r$

$\frac{4!}{(4-3)!}$ = 24 ways of selecting and ordering 3 or 4 letters, but only 4 ways if order does not matter.

ABCD as a Permutation Table

Combinations are the number of ways selecting $r$ items from a group of $n$ items where order does not matter.
To take out all the ways $r$ can happen, we divide out all the ways $r!$ can happen.
The number of combinitions of $n$ objects taken $r$ at a time is given by \(C\binom{n}{r} = C(n, r) = nCr = \frac{n!}{r!(n-r)!}\) $\implies$ Also called $n$ choose $k$, noted \binom{n}{k} = \frac{n!}{k!(n-k)!}$

Counting Subsets of an $n$-Set
Consider a binomial situation, where there is a yes or no, success or failure, possibility happening $n$ times. The number of ways this can happen is $2^n$. There are $2^n$ subsets of a set with $n$ objects.

Exercises

  1. How many ways can 8 runners in a track race finish?
  2. Subway is running a special on 6” subs for $3.99 with 6 choices of bread, 3 choices of meat, and 2 choices of cheese. Not including toppings or sause, how many choices are available?
  3. How many ways can 2 students of a class of 25 be selected for room reps?
  4. How many distinguishable ways can the letters, TOPPINGS be written?
  5. A license plate has 3 letters followed by 3 digits. How many possible plates are there?
  6. From a group fo 10 people in theater, how many possible ways can a teacher select roles for a show on the 7 dwarfs?
  7. In the original version of poker, straight poker, a five-card hand is dealt from a standard deck of 52. How many different hands are possible?
  8. A coin is flipped 10 times, and the sequence is recorded.
    1. How many sequences are possible?
    2. How many sequences have exactly 7 heads?
  9. A particular subway advertises 256 ways for a sandwich to be fixed with veggies. How many veggie toppings does this subway offer?
  10. Calculate $_{10}C_3$ without a calculator.