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Arrangements

Arrangements & combinations

Japanese school year: Elementary 6

What you learn

Learn how to find all possible ordered arrangements (permutations) of items systematically using tree diagrams and lists. Counting outcomes without omission or duplication forms the foundation for probability theory. Cultivating organized habits in listing possibilities will help you solve these combinatorial problems with ease.

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Key points

This formula calculates the factorial n!n!, the total number of ways to arrange nn distinct items in a line. Multiply all consecutive integers from nn down to 1.

n!=n(n−1)⋯2×1n! = n(n - 1)\dotsm 2 \times 1

This formula calculates the number of permutations when choosing and lining up rr items from nn distinct items. Multiply rr consecutive decreasing integers starting from nn.

nPr=n(n−1)⋯(n−r+1){}_n\mathrm{P}_r = n(n - 1)\dotsm(n - r + 1)

This formula expresses permutations using factorials. Divide the total ways to arrange all nn items, n!n!, by the ways to arrange the remaining unchosen (n−r)(n - r) items, (n−r)!(n - r)!.

nPr=n!(n−r)!{}_n\mathrm{P}_r = \frac{n!}{(n - r)!}

This is the multiplication rule for sequential events. If the first step has aa choices and each leads to bb subsequent choices, the total number of combinations is a×ba \times b.

Choose a set to practice.