Permutations count arrangements, so order matters. Combinations count groups, so order does not. The ACT signals which one with the wording. The probability that uses these counts is in probability problems.
Which word you heard
“How many ways can 5 students line up” is a permutation. First place is different from second. 5 × 4 × 3 × 2 × 1 = 120. “How many ways can you choose 3 of 5 students for a committee” is a combination. The committee{A, B, C} is the same group as {C, B, A}. Divide the permutation by the ways to order the chosen people: 5 × 4 × 3 / (3 × 2 × 1) = 10.
When you do not need the formula name
Filling different slots — a president, then a secretary — is still multiply-as-you-go, which is the counting principle. You only divide when the slots are identical. The broader topic is in statistics and probability. Overlapping groups are a different picture, in Venn diagrams.
The bottom line
Lineup or prize order: multiply and stop. Unordered group: multiply, then divide by the internal arrangements.
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