Choosing 3 people from 10 for a committee, or for president, vice-president and treasurer, use the same numbers but different arithmetic — because in one case order matters and in the other it does not.
How it works
Permutations and combinations
nPr = n! / (n − r)! · nCr = n! / (r! (n − r)!)
n is the total pool, r is how many are chosen. A combination is always the permutation count divided by r!, because every combination corresponds to r! different orderings that a permutation count treats as separate outcomes.
The only question that matters: does order matter?
Choosing 3 people from 10 for an unranked committee is a combination — the three names are the whole answer, whichever order they were picked in. Choosing the same 3 people from 10 for president, vice-president and treasurer is a permutation — the same three names assigned to different roles are three (or six, depending how many roles) genuinely different outcomes. Get this one question right and the rest of the arithmetic follows automatically.
How to use this calculator
- Enter the total pool size (n).
- Enter how many are being chosen (r).
- Read both figures — use permutations if order matters for your situation, combinations if it does not.
Frequently asked questions
Why is nCr always smaller than nPr for the same n and r?
Because nCr collapses every group of r people that could be arranged in different orders down to one count. Dividing by r! removes exactly the ordering information that made nPr larger in the first place.
What does nC0 = 1 actually mean?
There is exactly one way to choose nothing from any group — the empty selection. It is a genuine, if slightly abstract, answer rather than a special case to ignore.
How does a lottery use combinations?
A typical lottery draw does not care what order the numbers were drawn in — 1, 2, 3, 4, 5, 6 is the same winning combination as 6, 5, 4, 3, 2, 1 — so lottery odds are computed with combinations, not permutations, which is why the odds are usually “worse” (a bigger denominator) than people expect if they were thinking in terms of order.
Can r be larger than n?
No — you cannot choose or arrange more items than exist in the pool, so this calculator declines that input rather than returning a meaningless number.
Why doesn’t this calculator compute n! directly for large n?
Because factorials grow enormous very quickly — 52! is far too large for a computer’s standard number type to represent exactly. Building the permutation count as a direct product avoids ever forming the full factorial, which is what lets this work correctly even for fairly large n.
