Counting: permutations and combinations

Permutations count ordered arrangements and combinations count unordered selections; with equally likely outcomes, probability = favourable count ÷ total count.

{}_nP_k = \frac{n!}{(n-k)!}, \qquad \binom{n}{k} = {}_nC_k = \frac{n!}{k!\,(n-k)!}

Permutations (order matters) and combinations (order does not matter), without replacement.

$n!$: $n$ factorial: $n(n-1)\cdots 1$ , with $0! = 1$ .
$n,\ k$: Number of items available and number chosen.

n_1 \times n_2 \times \cdots \times n_r

Multiplication principle: number of ways to make r successive choices.

$n_j$: Number of options at step $j$ .

When to use

Probabilities with equally likely outcomes (cards, lotteries, random committees).
Building blocks of the binomial and hypergeometric distributions.
With replacement and order mattering: $n^k$ possibilities.

How to interpret

Ask two questions: does order matter (positions, rankings → permutations), and is selection with or without replacement? Then $P(\text{event}) = \text{favourable}/\text{total}$ , counted the same way in numerator and denominator.

Common mistakes

Counting ordered outcomes in the numerator and unordered ones in the denominator.
Using permutations for a committee (order irrelevant) or combinations for a podium (order relevant).
Applying "favourable/total" when outcomes are not equally likely.

Example

From 10 candidates: a chair, secretary and treasurer can be chosen in $_{10}P_3 = 720$ ways; a 3-person committee in $\binom{10}{3} = 120$ ways ( $720/3!$ ). There are $\binom{52}{5} = 2\,598\,960$ five-card hands.