NumberTally Permutation & Combination Calculator
Compute permutations (nPr) and combinations (nCr) side by side. Enter n and r, and get both answers instantly with the factorial working shown step by step — plus a growth table that shows how the numbers explode as n increases. Free, no sign-up.
Advanced options
Shows nPr and nCr for each n in the range, keeping your r fixed.
Permutations vs combinations
Permutations count arrangements where order matters — first, second, third place. Combinations count groups where order doesn't matter — a 3-person committee is the same no matter the order names are drawn. That's why nPr is always bigger: every combination of r items can be arranged in r! orders, so nPr = nCr × r!.
Why the numbers grow so fast
Factorials multiply every integer below n, so the counts explode: choosing 3 from 10 gives 720 permutations but only 120 combinations, and choosing 5 from 20 gives 1,860,480 permutations. The growth table below makes this concrete — watch the nPr column outrun nCr as n climbs.
Frequently Asked Questions
What is the difference between permutations and combinations?
Order matters for permutations (nPr), not for combinations (nCr). Picking a president, VP, and treasurer is a permutation; picking any 3 committee members is a combination.
What does nCr mean in probability?
It's the number of equally likely favorable groups. The probability of an event is (favorable nCr outcomes) ÷ (total nCr outcomes) — for example, lottery odds are 1 in nCr.
Why is r! in the denominator of the nCr formula?
Because each combination of r items can be ordered in r! ways, and combinations don't care about order — so you divide the permutations by r! to collapse those duplicates into one.
What happens if r is larger than n?
Both are zero — you can't choose more items than exist. The calculator flags this as an error; note that r = n gives 1 for combinations (exactly one way to take everything).
What is 0! and why does it equal 1?
By convention 0! = 1: there's exactly one way to arrange nothing. It also keeps the formulas consistent — nCn = n!/(0!·n!) = 1, as it should be.