Permutation & Combination Calculator — nPr and nCr Formulas

Here's the classic stumbling block: you're counting arrangements and you know whether order matters — but you're not sure which formula to reach for. Permutations and combinations are both about choosing from a group, with one key difference. Calculate both with the CalcHub Permutation & Combination Calculator.

The Core Difference

Permutation (nPr): Order matters. Choosing first, second, and third place is different from choosing third, first, and second. Combination (nCr): Order doesn't matter. Choosing teammates {Alice, Bob, Carol} is the same regardless of who you pick first.

A useful way to remember: your PIN code is a permutation (order matters). Your pizza toppings are a combination (doesn't matter what order they're listed).

Permutation Formula

$$P(n, r) = \frac{n!}{(n-r)!}$$

Where n is the total number of items and r is how many you're selecting.

Example: How many ways can 3 runners finish in 1st, 2nd, and 3rd from a field of 8?

P(8, 3) = 8! / (8−3)! = 8! / 5! = (8 × 7 × 6 × 5!) / 5! = 336 ways

Combination Formula

$$C(n, r) = \frac{n!}{r!(n-r)!} = \binom{n}{r}$$

Example: How many ways can you choose 3 people from a group of 8 to form a committee?

C(8, 3) = 8! / (3! × 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56 ways

The ratio P(8,3) / C(8,3) = 336 / 56 = 6 = 3! — exactly the number of ways to arrange 3 chosen items. This makes sense: every combination corresponds to 3! permutations.

Factorial Reference Table

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5,040
8	40,320
10	3,628,800
12	479,001,600

Note: 0! = 1 by convention — it's the empty product, meaning there's exactly one way to arrange nothing.

Real-World Examples

Lottery (Combination): India's lottery picks 6 numbers from 1–49. C(49, 6) = 49! / (6! × 43!) = 13,983,816 possible tickets. Order doesn't matter — matching the same 6 numbers wins regardless of draw order. Password (Permutation): A 4-digit PIN from digits 0–9, no repeats. P(10, 4) = 10!/6! = 10 × 9 × 8 × 7 = 5,040 combinations Card Hands (Combination): 5-card poker hand from 52 cards. C(52, 5) = 52! / (5! × 47!) = 2,598,960 possible hands Seating Arrangements (Permutation): 5 people in 5 chairs. P(5, 5) = 5! = 120 arrangements

Permutations with Repetition Allowed

If items can repeat: n^r

Example: 4-digit PIN allowing repeated digits: 10⁴ = 10,000 possible PINs (0000 through 9999)

How do I know if a problem is a permutation or combination?

Ask yourself: if I swap the order of my chosen items, do I get a different result? A different race finish — yes, it's a permutation. A different team roster — no, it's a combination. Phrases like "arrangements," "ranking," "sequence" suggest permutations. "Groups," "teams," "selections," "subsets" suggest combinations.

What does C(n,r) = C(n, n-r) mean?

Choosing 3 items from 10 is equivalent to choosing which 7 items to leave out. So C(10,3) = C(10,7) = 120. This symmetry is also reflected in Pascal's Triangle, where C(n,r) appears at row n, position r.

Why does 0! equal 1?

Mathematically, 0! = 1 makes combinatorics consistent: C(n, 0) = n! / (0! × n!) = 1, which correctly represents "there's exactly one way to choose nothing from a set." It also makes the gamma function and many series expansions work cleanly.

Related Calculators

• Probability Calculator
• Factorial via Exponent Calculator
• Prime Number Checker