nCr and nPr Calculator

Choosing 5 cards from 52 gives C(52, 5) = 2,598,960 possible poker hands, so any one hand has a 1 in 2,598,960 chance. When order matters, as in awarding first to fifth place among 52 entrants, P(52, 5) = 311,875,200. The fields open on n = 10 and r = 5: a committee of 5 picked from 10 people can be formed in C(10, 5) = 252 ways, and lining up 5 of the 10 in order gives P(10, 5) = 30,240. A 6-from-49 lottery has C(49, 6) = 13,983,816 possible tickets.

The answer is a count, not a probability. For a probability, divide the number of outcomes you want by it: one lottery ticket wins with a chance of 1 in 13,983,816. The answer follows n, r and the calculation type as you change them. A fractional value such as 5.7 is rejected with a message.

C(n, r) = n! ÷ (r! × (n − r)!) and P(n, r) = n! ÷ (n − r)!

n and r are whole numbers with 0 ≤ r ≤ n ≤ 170, and C(n, 0) and P(n, 0) are 1. The cap is 170 to keep the calculation and the printed steps short. The count itself is computed with exact integer arithmetic (a running product for P and a running product with exact division for C), so there is no rounding: C(49, 6) prints 13,983,816 and P(33, 11) prints 7,725,366,544,896,000. Counts up to 2⁵³ − 1 are shown in full with comma groups; larger ones are shown in exponent form with seven significant digits, such as C(100, 50) = 1.008913e+29, and Show Steps lists every digit (100891344545564193334812497256 for that one). Show Steps writes each factorial out in full for n up to 10, as in C(5, 2) = 5 × 4 × 3 × 2 × 1 / (2 × 1 × 3 × 2 × 1); above that it shows n! and the value of each factorial.

There are no options for repetition (choosing with replacement), circular arrangements or permutations of repeated items. The permutation and combination calculator covers repetition and n up to 1000.