Permutation & Combination Calculator: nPr and nCr

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Formula at a glance

nPr & nCr Calculator: Permutations — nPr = n! / (n − r)!. Example: Choose 3 from 10 → Permutations nPr: 720 — How the nPr & nCr Calculator works: Permutations — nPr = n! / (n − r)!

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📘 About the Permutation and Combination Calculator

Permutations count arrangements where order matters — race finishing positions, PIN codes, seating plans. Combinations count selections where order does not matter — lottery numbers, committees, pizza toppings. nPr = nCr × r!, because each selection can be arranged in r! ways.

Enter n and r and whether items can repeat. The calculator gives nPr and nCr exactly using big-number arithmetic (so 1000C500 shows all 300 digits), with the formulas and the counting rule behind each.

🧮 Formula

Permutations

nPr = n! / (n − r)!

Combinations

nCr = n! / (r!(n − r)!)

With repetition

nʳ (order matters), C(n + r − 1, r) (order doesn’t)

✏️ Worked examples

Choose 3 from 10

You enter
Total items n	10
Items chosen r	3
Repetition allowed?	No

You get
Permutations nPr	720
Combinations nCr	120
Order matters?	Permutations: yes (ABC ≠ CBA) · Combinations: no (ABC = CBA)
Check	nPr = nCr × r! (each group of 3 can be arranged in 3! ways)

nPr = 720, nCr = 120.

Lottery: 6 numbers from 49

You enter
Total items n	49
Items chosen r	6
Repetition allowed?	No

You get
Permutations nPr	10,068,347,520
Combinations nCr	13,983,816
Order matters?	Permutations: yes (ABC ≠ CBA) · Combinations: no (ABC = CBA)
Check	nPr = nCr × r! (each group of 6 can be arranged in 6! ways)

nCr = 13,983,816.

🧭 How to use this calculator

What you enter

Total items n
Items chosen r
Repetition allowed?

What you get

Permutations nPr
Combinations nCr
Order matters?
Check

❓ Frequently asked questions

What is the difference between permutation and combination?

Permutations count ordered arrangements (ABC ≠ BAC); combinations count unordered groups (ABC = BAC).

How many ways can I choose 6 numbers from 49?

49C6 = 13,983,816 — the odds of winning a 6/49 lottery jackpot with one ticket.

What is 0! and nC0?

0! = 1 by definition, so nC0 = nCn = 1 (there is one way to choose nothing or everything).

🔀 Permutation and Combination Calculator (nPr, nCr)