Combinations

When order doesn’t matter

A combination is a selection where order doesn’t matter. Choosing {Ana, Ben, Cara} for a committee is the same as {Cara, Ana, Ben}.

The formula

nCr = n! / [r!(n − r)!]

Also written C(n, r) or as a binomial coefficient "n choose r".

Example: Choose 3 students from 8 for a committee.

8C3 = 8!/(3! × 5!) = (8 × 7 × 6)/(3 × 2 × 1) = 56

Permutations vs combinations

8P3 = 336 ordered arrangements, but each group of 3 can be ordered 3! = 6 ways:

336 ÷ 6 = 56 ✓

So nCr = nPr ÷ r!

Useful facts

• nC0 = nCn = 1
• nC1 = n
• nCr = nC(n − r): choosing 3 to include is the same as choosing 5 to leave out (8C3 = 8C5)

Pascal’s triangle

The entries of Pascal’s triangle are combinations.

Row 5, position 2 (counting from 0) = 5C2 = 10. Each entry is the sum of the two above it.

Combinations with conditions

Example: A committee of 4 from 6 girls and 5 boys, with exactly 2 girls.

6C2 × 5C2 = 15 × 10 = 150

Example: "at least" — committee of 3 from 6 girls and 5 boys with at least 1 girl.

• Total: 11C3 = 165
• No girls: 5C3 = 10
• At least 1 girl: 165 − 10 = 155

Cards

5-card hands from a 52-card deck: 52C5 = 2,598,960.

P(all hearts) = 13C5 / 52C5 = 1,287/2,598,960 ≈ 0.000495.

Lotteries

Lotto 6/49: 49C6 = 13,983,816 possible tickets. One ticket’s chance of winning the jackpot: about 1 in 14 million.

Worked examples

Example 1: 10C2 = 45.

Example 2: Choose 2 toppings from 7. 21.

Example 3: How many handshakes among 12 people? 12C2 = 66.

Common mistakes

• Using permutations when order doesn’t matter.
• Adding when you should multiply (independent choices multiply).
• Forgetting the complement for "at least" problems.

Try it yourself

1. 7C3
2. 9C9
3. Choose a team of 5 from 12 players.
4. From 5 math and 4 science books, choose 2 of each.
5. A 3-person committee from 4 adults and 6 teens with at least 1 adult.

Answers: 1) 35 2) 1 3) 792 4) 60 5) 100 (120 − 20)

Quick recap

• A combination ignores order: nCr = n!/[r!(n − r)!].
• nCr = nPr ÷ r!; entries of Pascal’s triangle.
• Multiply independent choices; use the complement for at least.