A combination counts the number of ways to select $r$ items from $n$ distinct items when order does not matter — a committee, a hand of cards, a subset.

How it works

Start from all $^nP_r$ ordered arrangements, then divide out the $r!$ orderings within each chosen group that you no longer wish to distinguish.

Common problem types

1. Direct selection — plug into the formula.
2. With a condition — split into cases (e.g. "exactly 2 women") and multiply the sub-selections.
3. Find $n$ and $r$ — take ratios of consecutive values to kill the factorials.

Quick example

From 7 men and 4 women, form a committee of 5 with exactly 2 women.

1. Choose 2 women from 4: $^{4}C_{2}=6$.
2. Choose the remaining 3 members from the 7 men: $^{7}C_{3}=35$.
3. Both must happen, so multiply: $6\times 35=\textbf{210}$ ways. …