Combination

Statistics
definition

Also known as: selection

Grade 9-12

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

Definition

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)!}.

💡 Intuition

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

🎯 Core Idea

Combinations count unordered groups: C(n,r) = P(n,r)/r! because the r! orderings of the same group all count as one combination.

Example

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

Formula

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

Notation

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

🌟 Why It Matters

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

💭 Hint When Stuck

Ask: does the order of selection matter? If not, count permutations first and then divide by the number of rearrangements (r!).

Formal View

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

🚧 Common Stuck Point

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

⚠️ 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)

Frequently Asked Questions

What is Combination in Math?

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)!}.

What is the Combination formula?

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

When do you use Combination?

Ask: does the order of selection matter? If not, count permutations first and then divide by the number of rearrangements (r!).

Prerequisites

permutationfactorial

How Combination Connects to Other Ideas

To understand combination, you should first be comfortable with permutation and factorial. Once you have a solid grasp of combination, you can move on to binomial coefficient.

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Visualization

Static

Visual representation of Combination