Permutation & Combination Calculator

Permutations and combinations both count ways to select items from a group, but they differ in one critical way: permutations care about order (arrangement matters), combinations do not (only the selection matters).

Permutation formula (order matters): P(n, r) = n! / (n − r)!

Combination formula (order does NOT matter): C(n, r) = n! / (r! × (n − r)!)

Where:

• n = total number of items to choose from
• r = number of items chosen
• n! = factorial of n (n × (n−1) × (n−2) × … × 1)

Worked example — Permutation: How many ways can you arrange 3 books chosen from a shelf of 8? P(8, 3) = 8! / (8−3)! = 8! / 5! = (8 × 7 × 6) = 336 arrangements

Worked example — Combination: How many ways can you choose 3 students from a class of 25 for a committee? C(25, 3) = 25! / (3! × 22!) = (25 × 24 × 23) / (3 × 2 × 1) = 13,800 / 6 = 2,300 combinations

Key question to determine which to use: Does switching the order create a different outcome?

• Lock combination 4-7-2 vs 2-7-4: YES different → permutation (despite the name “combination lock”!)
• A 3-person committee with Alice, Bob, Carol vs Carol, Alice, Bob: same committee → combination

Special cases:

• All items chosen: P(n,n) = n! (arrangements of all items)
• Repetition allowed: n^r (passwords with repeated characters)
• Circular arrangements: (n−1)! (rotating a lineup doesn’t create a new one)

Real-world uses: Lottery odds (combinations), race finishing orders (permutations), password security (permutations with repetition), team selection (combinations).