Combination Formula

The Formula

C(n, r) = \frac{n!}{r!(n - r)!}

When to use: How many ways to choose a group? \{A, B, C\} = \{C, B, A\}.

Quick Example

Choose 2 from A, B, C: \{A, B\}, \{A, C\}, \{B, C\} = 3 ways.

Notation

C(n, r), _nC_r, or \binom{n}{r} all denote combinations of r items from n

What This Formula Means

A combination is an unordered selection of objects — the number of ways to choose r items from n distinct items is C(n,r) = \frac{n!}{r!(n-r)!}.

How many ways to choose a group? \{A, B, C\} = \{C, B, A\}.

Formal View

\binom{n}{r} = \frac{n!}{r!(n-r)!} for 0 \leq r \leq n, with \binom{n}{r} = \binom{n}{n-r}

Worked Examples

Example 1

easy
A committee of 3 is to be chosen from 8 people. How many different committees are possible?

Solution

  1. 1
    Recall the combination formula for unordered selections: \binom{n}{r} = \frac{n!}{r!(n-r)!}, with n = 8, r = 3.
  2. 2
    Cancel common factorial terms: \binom{8}{3} = \frac{8!}{3! \cdot 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1}
  3. 3
    Calculate: \frac{336}{6} = 56

Answer

\binom{8}{3} = 56
Combinations count selections where order does not matter. Choosing members A, B, C for a committee is the same as choosing C, B, A.

Example 2

medium
From a group of 6 men and 4 women, how many committees of 5 can be formed that include exactly 3 men and 2 women?

Common Mistakes

  • Using combinations when order matters — picking 1st, 2nd, and 3rd place requires permutations
  • Forgetting to divide by r! when converting from permutations to combinations
  • Confusing C(n, r) with C(r, n) — the larger number must be n (the pool), not r (the selection)

Why This Formula Matters

Combinations count lottery tickets, committee selections, and any situation where you pick a group and the order of picking does not matter.

Frequently Asked Questions

What is the Combination formula?

A combination is an unordered selection of objects — the number of ways to choose r items from n distinct items is C(n,r) = \frac{n!}{r!(n-r)!}.

How do you use the Combination formula?

How many ways to choose a group? \{A, B, C\} = \{C, B, A\}.

What do the symbols mean in the Combination formula?

C(n, r), _nC_r, or \binom{n}{r} all denote combinations of r items from n

Why is the Combination formula important in Math?

Combinations count lottery tickets, committee selections, and any situation where you pick a group and the order of picking does not matter.

What do students get wrong about Combination?

C(5, 2) = C(5, 3) = 10. Choosing 2 to include = choosing 3 to exclude.

What should I learn before the Combination formula?

Before studying the Combination formula, you should understand: permutation, factorial.

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