Permutations (nPr) Calculator

How it is calculated

The tool multiplies the first r terms of the falling product: nPr = n × (n−1) × (n−2) × … × (n−r+1), which equals n! ÷ (n−r)! without ever computing the full factorials. Here n is the number of distinct items available and r is how many you place in order; both must be whole numbers, 0 or larger, and r cannot exceed n. With r = 0 the answer is 1 (one way to arrange nothing). While the answer fits in a 64-bit integer (about 9.2 × 10¹⁸) every digit is exact, for example 20P20 = 2,432,902,008,176,640,000. Larger answers are shown in scientific notation with five significant digits, such as 8.0658 × 10⁶⁷ for 52P52. If the true value is beyond what a computer number can hold (about 1.8 × 10³⁰⁸), or the input is a decimal or has r greater than n, there is no result.

Common mistakes

• Using nPr when order does not matter. Picking a committee of 3 from 10 is a combination (120 ways), not nPr (720 ways).
• Swapping n and r: r is the number placed in order and cannot be bigger than n, so 3P10 returns nothing.
• Entering n as the number of positions and r as the number of items; n is always the pool you choose from.
• Treating repeated items as allowed. nPr assumes every item is distinct and used at most once; a PIN code where digits can repeat is 10⁴, not 10P4.
• Reading a result like 8.0658 × 10⁶⁷ as exact. Beyond about 9.2 × 10¹⁸ the tool gives only five significant digits, written as a mantissa times a power of ten.