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Permutation and Combination Calculator

Counting questions look simple until the order starts to matter. Picking three committee members from ten is one scenario; seating those three in ranked roles is another, and the second answer is six times the first. Combinations count selections where order is ignored, permutations count arrangements where it is not, and this calculator returns both from the same two numbers, keeping the gap visible at a glance.

Enter values

Size of the set — whole numbers 0 to 60

How many you pick or arrange — must not exceed n

Results update as you type.

Results

Combinations (nCr)
120
Permutations (nPr)
720

How is the Permutation and Combination calculated?

Enter two whole numbers: n, how many items are available, and r, how many you pick or arrange. The page computes nCr with a running product that multiplies and divides one step at a time — no factorial is ever formed, so even C(60, 30) returns promptly. Combinations lead the results because most real questions are selections, with permutations beside them. A reversed pair, where r is larger than n, has no counting answer and is reported as such.

Formula

nCr = n! ÷ (r! × (n − r)!), the ways to choose r items from n when order is ignored. nPr = n! ÷ (n − r)!, the ways to arrange r items drawn from n when order counts. n is the set size and r the number taken. Each is evaluated as a product of r factors — nPr = n × (n − 1) × … × (n − r + 1) — which keeps intermediate values small.

Example

A panel of 10 volunteers must send 3 to a conference. Ignoring order, the possible delegations number C(10,3) = (10 × 9 × 8) ÷ (3 × 2 × 1) = 120. Seating those 3 in ranked roles multiplies by their internal order: P(10,3) = 10 × 9 × 8 = 720, exactly 120 × 3!. Choosing 6 numbers from 50 in a lottery gives C(50,6) = 15,890,700 combinations — the figure printed beside every jackpot — while counting ordered draws would total 50 × 49 × 48 × 47 × 46 × 45 = 11,441,304,000.

Things to remember

When a question says combinations, read it as a group; seats, codes, rosters and races all fold order into the answer. Lotteries pay on combinations precisely because counting order would inflate the odds against you. Check the two rows against each other: permutations divided by combinations always leaves r factorial.

Frequently Asked Questions

What is the difference between a permutation and a combination?

A combination counts groups where picking order is ignored, while a permutation counts full arrangements where a different order means a different outcome. Choosing 3 of 10 gives 120 combinations; arranging those 3 gives 720 permutations, and the ratio between the two is always 3! = 6.

How do you calculate n choose r by hand?

Write nCr as r factors running down from n, then divide by r factorial. For 10 choose 3 that is (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120. Cancelling pairs before multiplying keeps the numbers small enough for mental arithmetic.

What is 50 choose 6 in a lottery?

15,890,700. Lottery combinations are exactly C(50,6), and the calculator reproduces that figure with the same product loop it applies to any other pair. Pay-outs deliberately ignore the order the numbers are drawn in — counting order would multiply the odds against you by 720.

Why can't the value of r exceed n?

You cannot choose or arrange more items than exist, so nCr and nPr are undefined whenever r exceeds n — no arrangement of 5 items can be drawn from a set of 3. The calculator flags the request plainly instead of returning a misleading figure.

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