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) = n!/r!(n-r)!.

The Formula

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

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

Quick Example

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

Notation

C(n,r)C(n, r), nCr_nC_r, or (nr)\binom{n}{r} all denote combinations of rr items from nn

What This Formula Means

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

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

Formal View

(nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n-r)!} for 0≤r≤n0 \leq r \leq n, with (nr)=(nn−r)\binom{n}{r} = \binom{n}{n-r}

Worked Examples

Example 1

easy
A committee of 33 is to be chosen from 88 people. How many different committees are possible?

Answer

(83)=56\binom{8}{3} = 56

First step

1
Recall the combination formula for unordered selections: (nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n-r)!}, with n=8n = 8, r=3r = 3.

Full solution

  1. 2
    Cancel common factorial terms: (83)=8!3!⋅5!=8×7×63×2×1\binom{8}{3} = \frac{8!}{3! \cdot 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1}
  2. 3
    Calculate: 3366=56\frac{336}{6} = 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 66 men and 44 women, how many committees of 55 can be formed that include exactly 33 men and 22 women?

Example 3

easy
A jar holds 5 different fruits. How many distinct fruit salads of 3 different fruits can be made? (Order does not matter inside a salad.)

Common Mistakes

  • Forgetting to divide by r!r! — that leaves a permutation count, which overcounts the orderings.
  • Using a combination when order matters — assignments and rankings need a permutation.
  • Mixing up rr and n−rn-r — note C(n,r)=C(n,n−r)C(n,r)=C(n,n-r), so choosing 3 to keep equals choosing the others to drop.

Why This Formula Matters

Combinations are the 'order doesn't matter' half of counting, and the dividing-by-r!r! step is exactly what prevents the overcounting that permutations would cause. They are the engine behind the binomial coefficient and Pascal's triangle. Recognizing it by "Does rearranging the chosen items leave it the same selection?" — rather than by familiar numbers — is what lets a student tell it apart from permutation and counting principle and binomial coefficient in a mixed problem set.

Frequently Asked Questions

What is the Combination formula?

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

What do the symbols mean in the Combination formula?

C(n,r)C(n, r), nCr_nC_r, or (nr)\binom{n}{r} all denote combinations of rr items from nn

Why is the Combination formula important in Math?

Combinations are the 'order doesn't matter' half of counting, and the dividing-by-r!r! step is exactly what prevents the overcounting that permutations would cause. They are the engine behind the binomial coefficient and Pascal's triangle. Recognizing it by "Does rearranging the chosen items leave it the same selection?" — rather than by familiar numbers — is what lets a student tell it apart from permutation and counting principle and binomial coefficient in a mixed problem set.

What do students get wrong about Combination?

The procedure for combination is the easy part; the trap is forgetting to divide by r!r!. Asking "Does rearranging the chosen items leave it the same selection?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Combination formula?

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

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