What is the difference between permutations and combinations?

Permutations count ordered arrangements - the order of selection matters. Combinations count unordered selections - the order does not matter. For example, choosing 2 letters from {A, B, C}: AB and BA are different permutations but the same combination.

What is the permutation formula?

P(n, r) = n! / (n − r)!. This counts the number of ways to arrange r items chosen from n distinct items.

What is the combination formula?

C(n, r) = n! / (r! × (n − r)!). Also written as "n choose r". This counts the number of ways to choose r items from n items without regard to order.

What does r = 0 mean?

Both P(n, 0) and C(n, 0) equal 1. There is exactly one way to arrange or choose nothing.

Permutations & Combinations Calculator - nPr and nCr

Permutations vs Combinations
Item	Value
P(10,3)	720
C(10,3)	120

Permutations vs Combinations

Item

Value

P(10,3)

720

C(10,3)

120

Permutations vs. combinations

Permutations P(n, r) count ordered arrangements: picking a president, vice president, and secretary from 10 candidates is a permutation because the roles are distinct. Combinations C(n, r) count unordered selections: choosing 3 players from 10 for a team is a combination because the order of selection doesn't matter.

Formulas

P(n, r) = n! / (n − r)!: multiply n down to (n − r + 1).

C(n, r) = n! / (r! × (n − r)!): also written as the binomial coefficient "n choose r".

Example: poker hands

A standard 5-card hand dealt from a 52-card deck: C(52, 5) = 2,598,960. Since the cards in a hand are unordered, combinations are used, not permutations.

Real-world problems guide

Scenario	Ordered?	Use
Selecting a committee of 3 from 10 people	No	C(10, 3) = 120
Arranging 4 books on a shelf from 8 choices	Yes	P(8, 4) = 1,680
4-digit PIN from digits 0–9 (no repeats)	Yes	P(10, 4) = 5,040
Lottery: pick 6 numbers from 49	No	C(49, 6) = 13,983,816
How many ways to rank 1st, 2nd, 3rd from 20 runners?	Yes	P(20, 3) = 6,840

Combinations with repetition

When repetition is allowed (you can choose the same item more than once) and order doesn't matter, the count is the "multiset coefficient": C(n + r − 1, r). For example, choosing 3 flavors from 5 options with repetition allowed = C(5 + 3 − 1, 3) = C(7, 3) = 35. This covers scenarios like choosing 3 scoops of ice cream where you can pick the same flavor multiple times.

Why n! grows so fast

Factorial growth is faster than exponential growth. 10! = 3,628,800. 20! ≈ 2.4 × 10¹⁸ - larger than the number of seconds in the age of the universe. 52! (the number of ways to shuffle a standard deck of cards) ≈ 8 × 10⁶⁷ - more than the number of atoms in the observable universe. This is why brute-force search of large combinatorial spaces is computationally infeasible.