choose() — Combinations

Counts the ways to pick k things from n when the order does not matter.

choose(n, k)

Key idea

• C(n,k), said "n choose k", counts ways to pick k items from n when ORDER DOES NOT matter. (e.g. C(5,2) = 10)
• Picking none has exactly one way, so C(n,0) always equals 1. (e.g. C(7,0) = 1)
• It is symmetric: choosing k to keep is the same as choosing n-k to drop. (e.g. C(5,2) = C(5,3) = 10)
• Formula: C(n,k) = n! / (k! x (n-k)!), where n! is the factorial of n. (e.g. C(5,2) = 120/(2x6) = 10)
• Use it for combinations like lottery picks or forming a team. (e.g. pick 3 of 5 -> C(5,3) = 10)

Definition

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$$

$n$

total items

$k$

how many you pick

$\left(\genfrac{}{}{0ex}{}{n}{k}\right)$

ways to choose, order ignored

$$\left(\genfrac{}{}{0ex}{}{5}{2}\right)=\frac{5!}{2!3!}=\frac{120}{2\cdot 6}=10$$

Where you'd use it

• A coach picks 3 players from a squad of 10 for a relay, and the order does not matter. choose(10, 3) = 120, so there are 120 possible teams.
• You want to know how many different 5-card poker hands a 52-card deck holds. choose(52, 5) ≈ 2,598,960 distinct hands.
• For a lottery where you tick 6 numbers out of 49, the number of possible tickets is choose(49, 6) ≈ 13,983,816, which is why your odds of winning are so tiny.

Frequently asked questions

choose vs permutations?

choose counts groups where order doesn’t matter (combinations); permutations count ordered arrangements.

What is choose(n, 0)?

1 — there’s exactly one way to choose nothing.