Counting Principles

Permutations nPr

Compute P(n, r) = n! / (n-r)!, the number of ordered arrangements of r items chosen from n distinct items.

Order matters in permutations.

permutations-nPr P(n, r) = n! / (n - r)! = n · (n-1) · ⋯ · (n-r+1)

Example

Use the permutation formula for ordered arrangements.

highlighted = computed this step

Step 1 — Set up

Identify n and r for the permutation.

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

Step 2 — Formula

Substitute into the permutation formula.

P(7,3)=7!4!P( 7 , 3 )=\frac{ 7 !}{ 4 !}

Step 3 — Cancel factorials

Cancel factorials to get the product.

7⋅6⋅5=2107 \cdot 6 \cdot 5 = \hl{210}

Step 4 — Result

State the permutation count.

P(7,3)=210P( 7 , 3 )= \hl{210}