Permutations & Combinations Worksheets

What it is

A factorial n! multiplies every whole number from n down to 1 (5! = 5 × 4 × 3 × 2 × 1 = 120). A permutation ₙPᵣ counts arrangements of r items from n where order matters: ₙPᵣ = n! ÷ (n − r)!. The (n − r)! matches the bottom of n!, so those like factors cancel and only r survive, counting down from n. A combination ₙCᵣ = ₙPᵣ ÷ r! removes the r! repeated orderings.

Why it matters

Counting techniques are the foundation of probability and statistics and appear across science, computing, and everyday decisions.

How to do it

1. Decide whether order matters (permutation, P) or not (combination, C).
2. Write ₙPᵣ as the fraction n! ÷ (n − r)!.
3. Cancel the (n − r)! tail shared by top and bottom, then multiply the r numbers left, counting down from n.
4. For a combination, divide that result by r!: ₙCᵣ = ₙPᵣ ÷ r!.

Examples by level

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

• Using a permutation when order doesn't matter (or vice versa).
• Dividing by the wrong factorial in a combination.
• Miscounting the number of factors in ₙPᵣ.

Tips

• Order matters → permutation; order doesn't → combination.
• The (n − r)! tail always cancels, so you never multiply all of n! by hand, and ₙCᵣ is always smaller than ₙPᵣ.

Key vocabulary

factorial (n!)

the product of every whole number from n down to 1

permutation

an arrangement where order matters; ₙPᵣ = n! ÷ (n − r)!

combination

a selection where order does not matter; ₙCᵣ = ₙPᵣ ÷ r!

Frequently asked questions

What is the difference between a permutation and a combination?

In a permutation the order matters; in a combination it does not. ₙCᵣ = ₙPᵣ ÷ r!.

How do you calculate ₙPᵣ?

Multiply r numbers starting from n and counting down: ₅P₃ = 5 × 4 × 3 = 60.