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About this tool
nPr and nCr for choosing r items from n.
The Permutation & Combination Calculator returns both permutations and combinations for choosing r items from n, using nPr = n! ÷ (n − r)! when order matters and nCr = n! ÷ (r! × (n − r)!) when it does not. Enter the two whole numbers and it shows each count side by side, so you can see how much the ordering assumption changes the answer. Aimed at students working through counting problems and anyone sizing up how many arrangements or selections actually exist.
Open Permutation & Combination Calculator on AltFTool — it loads instantly in your browser.
Enter the size of your set in the Total items (n) field, which starts at 6; decimal values are rounded down before the calculation.
Enter how many you are picking in the Chosen (r) field, which starts at 3 and also rounds decimals down; r has to be between 0 and n, otherwise the result reads Need 0 ≤ r ≤ n.
There is no permutations-or-combinations switch — both are computed live as you type: the headline gives the nCr total followed by the word combinations, and the two cards below it are labelled Permutations (nPr) and Combinations (nCr).
nPr and nCr are shown together, so you can immediately see the r! factor separating them.
It refuses r greater than n and negative inputs rather than returning a meaningless number, and floors non-integer entries.
Seeing 336 permutations against 56 combinations for 8 choose 3 shows exactly what 'order matters' costs.
A permutation counts arrangements where order matters; a combination counts selections where it does not. They differ by a factor of r!: choosing 3 from 6 gives 120 permutations but 20 combinations, because each group of 3 can be arranged 3! = 6 ways.
Divide n! by r! × (n − r)!. For 49 choose 6 that is 49! ÷ (6! × 43!) = 13,983,816 — the number of possible tickets in a 6-from-49 lottery. In practice you cancel most of the factorial instead of expanding it: 49×48×47×46×45×44 ÷ 720.
Because you cannot choose more items than exist — (n − r)! would need the factorial of a negative number, which is undefined. The calculator shows a dash and the condition 0 ≤ r ≤ n instead of a result.
Only up to a point. The factorials are computed in double-precision floating point, which is exact through 18! and then starts rounding, and anything from 171! upwards overflows to infinity. For counting problems in the tens, results are exact; for very large n treat them as approximations.
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