Permutations & Combinations Calculator

How it is calculated

Both counts select r items out of n distinct items. Permutations count ordered selections: nPr = n! ÷ (n − r)! = n × (n−1) × … × (n−r+1). Combinations ignore order, so they divide by the r! ways to reorder the same group: nCr = nPr ÷ r!. The tool multiplies only the r needed terms rather than computing full factorials, and rounds n and r to whole numbers first. It returns nothing if n or r is negative, or if r is larger than n. Items are distinct and cannot repeat; selections with repetition use different formulas. Very large results are shown rounded because they exceed exact integer precision.

Common mistakes

• Using combinations for a race podium or a PIN. When first, second and third place differ, order matters, so you need nPr, not nCr.
• Using permutations for a lottery or a committee, where only the group matters. That overcounts by a factor of r!.
• Entering r larger than n. You cannot pick 5 different items from 3, so the tool gives no result.
• Applying the formulas when items can repeat, like a 4-digit code where digits may be reused. That count is simply 10⁴, not nPr.