Counting Principles

Combinations nCr

Compute C(n, r) = n! / (r! · (n-r)!), the number of unordered selections of r items from n distinct items.

Order does not matter in combinations.

combinations-nCr C(n, r) = n! / (r! · (n - r)!) = P(n, r) / r!

Example

Use the combination formula for unordered selections.

highlighted = computed this step

Step 1 — Set up

Identify n and r for the combination.

n=7r=3n= 7 \quad r= 3

Step 2 — Formula

Substitute into the combination formula.

C(7,3)=7!3!⋅4!C( 7 , 3 )=\frac{ 7 !}{ 3 !\cdot 4 !}

Step 3 — Numerator

Compute the ordered numerator.

7⋅6⋅5=2107 \cdot 6 \cdot 5 = \hl{210}

Step 4 — Denominator

Compute r factorial in the denominator.

3!=63 != \hl{6}

Step 5 — Divide

Divide by r factorial because order does not matter.

210/6=35210 / 6 = \hl{35}