Combinations: formula, calculator and worked example

What the symbols mean

$C$: the number of selections, "n choose r", also written $_nCᵣ$
$n$: the number of items to choose from
$r$: how many are chosen (r ≤ n)

When to use it

Use combinations when only *which* items are chosen matters, not their order: poker hands, lottery draws, teams, pizza toppings. They also count the ways to get k successes in n trials, which is why they appear in the binomial distribution and the binomial theorem.

Worked example

How many different 5-card hands can be dealt from a standard 52-card deck? (n = 52, r = 5)

Start from the formula.
$(\binom{n}{r}) = \frac{n!}{r! (n - r)!}$
Cancel the larger factorial: the top keeps 5 factors and the bottom is 5!.
$(\binom{52}{5}) = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2598960$

Answer 2,598,960 possible hands

Why it works

Count ordered arrangements first, $n!/(n − r)!$ , then divide by the $r!$ orders in which the same r items could have been picked, since each selection was counted once per order.

Common mistakes

Using permutations for unordered selections, which overcounts by a factor of $r!$ .
Forgetting the symmetry $C(n, r) = C(n, n − r)$ , which makes $C(100, 98)$ easy: it equals $C(100, 2) = 4950$ .
Adding instead of multiplying for choices made together: 3 of 10 men and 2 of 8 women is $C(10, 3)×C(8, 2)$ .

Calculator

What the symbols mean

When to use it

Worked example

Why it works

Common mistakes