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Permutations vs Combinations
Permutations count arrangements where order matters (e.g., race placements, passwords). Combinations count selections where order does not matter (e.g., lottery numbers, committee members). The key question is: does rearranging the same items create a different outcome? If yes, use permutations; if no, use combinations.
How to Calculate Permutations (nPr)
The permutation formula is nPr = n! / (n-r)!. For example, arranging 3 items from 10: 10P3 = 10! / 7! = 10 × 9 × 8 = 720. This counts every possible ordered arrangement of r items chosen from n total items.
How to Calculate Combinations (nCr)
The combination formula is nCr = n! / (r! × (n-r)!). For example, choosing 3 items from 10: 10C3 = 10! / (3! × 7!) = 120. This is always less than or equal to the permutation count because it eliminates duplicate arrangements of the same items.
Real-World Applications
Permutations apply to PIN codes, seating arrangements, race outcomes, and any scenario where sequence matters. Combinations apply to lottery odds, card hands, team selection, and survey sampling. Understanding both is essential for probability calculations and counting problems.
Frequently Asked Questions
A factorial (n!) is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow extremely fast — 20! is over 2.4 quintillion.
No. You cannot choose or arrange more items than are available. If r > n, both nPr and nCr are 0 (impossible).
nPr = nCr × r!. In other words, permutations equal combinations multiplied by the number of ways to arrange r items. This makes sense because permutations count each arrangement separately, while combinations do not.
Combinations with repetition use the formula (n+r-1)! / (r! × (n-1)!). This applies when the same item can be chosen more than once, like scoops of ice cream from multiple flavors.