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Permutations

Counting & probability

Japanese school year: Math A

What you learn

Learn how to calculate permutations to find the number of ways to arrange items in a specific order. This mathematical technique is widely used in problems involving passcode combinations, scheduling order, and ranking possibilities. Prior knowledge of the multiplication principle and tree diagrams will help you master these permutation formulas.

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Key points

This formula calculates the total number of permutations when choosing rr distinct items from nn and arranging them in order. Use it when the sequence of selection matters.

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

This defines the factorial n!n!, which gives the total number of ways to arrange all nn distinct items in a line. It multiplies all positive integers from nn down to 1.

n!=n(n−1)⋯2⋅1n! = n(n - 1)\cdots 2 \cdot 1

These mathematical conventions define 0!=10! = 1 and nP0=1{}_n\mathrm{P}_0 = 1. They ensure that permutation formulas work consistently even when r=nr = n or r=0r = 0.

0!=1,nP0=10! = 1, \quad {}_n\mathrm{P}_0 = 1

Choose a set to practice.