Factorials, Permutations and Combinations: The Difference That Matters

Order matters for permutations. Order doesn't matter for combinations. That single distinction is the whole game.

Google this topic and you will get a hundred answers, half of them wrong. Here is the version that holds up. Order matters for permutations. Order doesn't matter for combinations. That single distinction is the whole game.

Factorial

n! = n × (n−1) × ... × 1. 5! = 120. 0! = 1 by convention.

Permutation P(n,r)

Ways to arrange r items from n where order matters. n!/(n−r)!

Combination C(n,r)

Ways to choose r items from n where order doesn't matter. n!/((n−r)!r!)

Pro tips

  • Lottery odds are combinations — 6/49 = C(49,6) = 13,983,816 ways.
  • Calculator nCr and nPr buttons handle the math for you — but understand what they compute.

Frequently asked questions

Why is 0! = 1? Convention that makes the combination formula work when r = n.

What about repeats? Different formulas — multinomial coefficients handle repeated items.

Bookmark the calculator linked above — you will need it more often than you think.

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