Permutation Calculator (nPr)
Calculate the number of permutations of n items taken r at a time (nPr).
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How to use this tool
- Enter n (total items) and r (items chosen) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your npr and the full breakdown beneath it.
nPr = n! / (n−r)! counts ordered arrangements of r items from n.
Formula
nPr = n! ÷ (n − r)! = n × (n−1) × (n−2) × … × (n−r+1)
Computed as a descending product: result = n × (n−1) × … × (n−r+1) (r terms).
How it works
This calculator finds the number of ordered arrangements (permutations) of r items selected from a set of n distinct items. Order matters: choosing A then B is counted separately from choosing B then A. The calculation multiplies r consecutive integers starting at n and descending, avoiding the need to compute full factorials. If r exceeds n, the result is 0 because you cannot arrange more items than exist.
Worked example
- n = 5 (total items), r = 2 (items chosen).
- List r = 2 descending integers starting at n = 5: 5, 4.
- Multiply: 5 × 4 = 20.
nPr = 20 ordered arrangements.
Common mistakes to avoid
- Confusing nPr with nCr; permutations count ordered arrangements so nPr >= nCr for any given n and r.
- Setting r greater than n, which is undefined; you cannot arrange more items than you have.
- Using nPr when the problem involves choosing without regard to order, e.g. selecting a committee where roles are not assigned.
Key terms
- Permutation
- An ordered arrangement of items where the sequence matters. Swapping any two items produces a different permutation.
- n (total items)
- The size of the pool from which items are drawn.
- r (items chosen)
- How many items are selected from the pool for each arrangement.
- Factorial (n!)
- The product of all positive integers from 1 to n. For example, 5! = 120. Used as the theoretical basis of nPr = n! ÷ (n−r)!.
- nPr vs nCr
- nPr counts ordered arrangements; nCr counts unordered groupings. nPr is always ≥ nCr for the same n and r.
Frequently asked questions
- What is a permutation?
- A permutation is an ordered arrangement. nPr = n!/(n-r)!. For example, P(5,2) = 5×4 = 20 ordered pairs from 5 items.