Combinations: Choosing Without Order
Big Idea: A combination is a selection of items from a group where the order does not matter. Choosing {A, B, C} is the same as choosing {C, A, B}.
🔀 Permutations (Order Matters)
Selecting a president, vice-president, and secretary from a group.
ABC ≠ BAC ≠ CAB
These are different arrangements.
🤝 Combinations (Order Doesn't Matter)
Selecting a committee of 3 people from a group.
{A, B, C} = {B, A, C} = {C, A, B}
These are all the same committee.
📐 The Combination Formula
The number of ways to choose $r$ items from $n$ items (without regard to order) is:
$$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$$
Where:
- $n$ = total number of items available
- $r$ = number of items being chosen
- $n!$ = $n$ factorial = $n \times (n-1) \times (n-2) \times \cdots \times 1$
- $\binom{n}{r}$ is read as "n choose r"