Factorials

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The factorial of a number $n$, denoted $n!$, is the product of all positive integers less than or equal to $n$.

$$ n! = \prod_{i = 0}^{n -1} (n - i) = n \times (n - 1) \times (n - 2) \times \ldots \times1 $$

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Example:

$5! = 5 × 4 × 3 × 2 × 1 = 120$

Factorials are the building blocks of permutations and combinations.

Permutations

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A permutation refers to an arrangement of objects where order matters.

Formula (when selecting $r$ objects from $n$ total):

$$ P(n, r) = \frac{n!}{(n - r)!} $$

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Example:
- How many ways can you arrange 3 letters out of 5 distinct letters (A, B, C, D, E)?

If some objects are repeated, the formula is modified to account for the indistinguishable items:

Formula (where $n_1, n_2, \ldots, n_k$ represent the number of repetitions for each object):

$$ P(n; n_1, n_2, ..., n_k) = \frac{n1}{n_1! n_2! ... n_k!} $$
Example:
- How many distinct ways can the letters in "AAB" be arranged?

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A combination refers to selecting objects where order does not matter.

$$ C(n, r) = {n \choose r} = \frac{n!}{r!(n - r)!} $$

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Example:
- How many ways can you choose 3 fruits from 5 distinct fruits (apple, banana, cherry, date, elderberry)?

When items can be selected more than once, we use the following formula:

Formula:

$$ C(n + r - 1, r) = \frac{(n + r - 1)!}{r! (n - 1)!} $$
Example:
- How many ways can you select 3 scoops of ice cream from 5 flavours if repeats are allowed?