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Combinations & Permutations (nCr, nPr)

How do I calculate nCr and nPr?

Result

120
720
3,628,800

How it is calculated

Choose r items from n. Permutations count ordered selections: nPr = n × (n−1) × … × (n−r+1). Combinations ignore order: nCr = nPr ÷ r!. The factorial shown is n! = 1 × 2 × … × n. Both n and r are whole numbers (decimals are cut off), r cannot exceed n and neither can be negative; n above 1,000,000 also returns no result. nCr, nPr and n! are exact while they fit in 64-bit integers; beyond that they are approximate floating-point values. n! shows "—" when n is 171 or more because it exceeds the float range, while nCr and nPr are still shown (n = 200, r = 2 gives 19,900 and 39,800).

Worked example

Pick 3 people out of 10. nPr = 10 × 9 × 8 = 720 ordered arrangements. Each group of 3 can be ordered in 3! = 6 ways, so nCr = 720 ÷ 6 = 120 distinct groups. The factorial 10! = 3,628,800.

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Common mistakes

  • Using permutations when order does not matter (a lottery draw is a combination) or the reverse (a podium finish is a permutation).
  • Reading the factorial output as n!/r!; it is simply n! and does not depend on r.
  • Entering r larger than n, which has no valid answer because you cannot pick more items than exist.
  • Typing a decimal such as 5.7 and expecting rounding; the value is truncated to 5.
  • Assuming repetition is allowed; these formulas pick each item at most once.

About this calculator

This calculator computes combinations (nCr), permutations (nPr) and the factorial (n!) from two inputs, n (the total number of items) and r (how many are being chosen or arranged). Permutations count arrangements where order matters, combinations count selections where order doesn't, and both formulas are built from the factorial function, which itself multiplies every whole number from 1 up to n.

These formulas are core to probability and statistics coursework, where students calculate the odds of a specific hand, drawing, or arrangement occurring out of all possible outcomes. Outside the classroom, the same math shows up whenever someone needs to know how many possible outcomes exist — the number of ways to arrange a race's finishing order, the number of possible lottery ticket combinations, or how many different teams of a given size can be formed from a larger group — situations that come up in game design, scheduling and odds calculation alike.

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Comments (1)

  • Editorial team

    Use nCr when the order does not matter and nPr when it does, with no repeats allowed in either. A familiar case is a lottery that draws 6 numbers from 49, where nCr gives 13,983,816 possible tickets. Results are exact while they fit in a 64-bit integer and approximate beyond that. For n of 171 or more the factorial shows a dash because it is out of range, yet nCr and nPr still appear. What kind of selection problem are you solving?

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