Permutations & Combinations

Calculate nPr and nCr for counting problems

Grades HS

n (total items, 1–20)

r (chosen, 0–n)

Factorial Expansions

5! = 5 × 4 × 3 × 2 × 1 = 120

2! = 2 × 1 = 2

3! = 3 × 2 × 1 = 6

Permutations — nPr

Order matters — arrangements

5P2 = 5! / (5 − 2)! = 5! / 3!

= 120 / 6 = 20

How many ways to arrange 2 books from 5? → 20

Combinations — nCr

Order doesn't matter — selections

5C2 = 5! / (2! × 3!)

= 120 / (2 × 6) = 10

How many ways to choose 2 toppings from 5? → 10

nCr is always ≤ nPr because combinations ignore order. Here 20 / 10 = 2 (= 2! = 2).

Pascal's Triangle(rows 0–5)

Highlighted cell: row 5, position 2 = 10

121

1331

14641

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