Combinatorics Calculator
This free permutation and combination calculator finds nPr (permutations, where order matters) and nCr (combinations, where order doesn't) from a total number of items (n) and how many are chosen (r). Nothing you enter is sent anywhere or stored.
How to Use It
- Enter n, the total number of items.
- Enter r, how many of them you are choosing or arranging.
- Permutations (nPr) and combinations (nCr) update instantly, no submit button.
How It Works
Permutations and combinations both answer "how many ways can I choose r items from a total of n," but they disagree on one key point: whether the order of the chosen items counts as a different outcome. Permutations count every distinct order as its own separate result (1st-2nd-3rd place is different from 2nd-1st-3rd); combinations only count which items were chosen, ignoring order entirely (the same 3 committee members are the same committee no matter which order they joined in).
Formulas
nPr = n! ÷ (n − r)!
nCr = n! ÷ (r! × (n − r)!)
A worked example: choosing 3 items from 5 (n=5, r=3) gives 5! ÷ 2! = 60 permutations, but only 5! ÷ (3! × 2!) = 10 combinations, since each group of 3 can be arranged 6 different ways. A 5 card poker hand from a 52 card deck (n=52, r=5) gives 311,875,200 permutations, but just 2,598,960 combinations, the actual number of distinct possible poker hands, since the order the 5 cards were dealt in doesn't change which hand you're holding.
Everything runs client-side in your browser. Nothing you enter is sent anywhere or stored.
Frequently Asked Questions
What do "n" and "r" mean?
n is the total number of items available to choose from, and r is how many of them you are actually choosing or arranging. For example, choosing a 5 card poker hand from a standard 52 card deck is n = 52, r = 5.
Why is the number of permutations always greater than or equal to the number of combinations?
Every combination (a group of r items, order ignored) corresponds to multiple permutations (the same r items, arranged in every possible order), so permutations always count at least as many outcomes as combinations for the same n and r. They are only equal when r is 0 or 1, since there is only one possible order for zero or one item.
Why can't r be greater than n?
You can't choose or arrange more items than actually exist to choose from, so r greater than n has no defined answer, the same way you can't deal 10 cards from a 5 card deck.
What is a real-world example of a permutation?
Arranging runners on a podium (1st, 2nd, 3rd place) is a permutation, since swapping which runner gets which place produces a genuinely different, distinct outcome. Order matters: gold-silver-bronze is a different result from silver-gold-bronze even with the same three runners.
What is a real-world example of a combination?
Choosing a 3 person committee from a group of 10 people is a combination, since the committee is the same group regardless of the order the members were picked in. A lottery draw and a poker hand work the same way: only which items end up selected matters, not the order they were dealt or drawn in.