Combinations

Definition

Combinations refer to the selection of items from a larger set where order does not matter. They are used to determine how many ways a subset of items can be chosen from the entire set without regard to the sequence of selection.

5 Must Know Facts For Your Next Test

1. The formula for combinations is C(n, k) = n! / [k!(n - k)!], where n is the total number of items and k is the number of items to choose.
2. Combinations are different from permutations because permutations consider order while combinations do not.
3. In combinations, C(n, k) is equal to C(n, n - k), meaning choosing k items from n is the same as choosing (n - k) items from n.
4. A combination problem often uses phrases like 'selecting', 'choosing', or 'picking' without concern for order.
5. Binomial coefficients, often found in binomial expansions, are calculated using combinations.

Review Questions

• What distinguishes a combination from a permutation?
• How would you calculate the number of ways to choose 3 items out of a set of 7?
• Explain why C(10, 2) equals C(10, 8).

Related terms

Permutation:

An arrangement or listing where order matters.

Factorial: The product of all positive integers up to a given number; denoted by n!.

Binomial Coefficient:

A component in binomial expansion that represents combinations and is written as (n choose k).