Combinations and permutations

Combinations or permutations? It all comes down to order

There is only one key question: does order matter? If it does, you want permutations: the podium of a race with 10 athletes has 10 × 9 × 8 = 720 possible outcomes (nPr, i.e. n! / (n−k)!), because gold-silver-bronze is not the same as bronze-silver-gold. If order doesn't matter, you want combinations: a classic 6/49 lottery ticket picks 6 numbers out of 49, and the possible draws are C(49,6) = 13,983,816, because playing 1-2-3-4-5-6 or 6-5-4-3-2-1 is exactly the same bet. nCr is always nPr divided by k! — the internal orderings that "collapse" into a single selection.

What about repetition? PINs, license plates and ice cream

With repetition, the same item can be used more than once. A 4-digit PIN allows repeated digits: 10⁴ = 10,000 permutations with repetition. A cup with 3 scoops chosen from 12 flavors — doubles and triples allowed — gives C(12+3−1, 3) = 364 combinations with repetition. These numbers blow up fast: 52! alone (the ways to shuffle a deck of cards) has 68 digits. That's why this tool shows the exact value and, once it passes 12 digits, also the scientific notation so you can read it at a glance.