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All courses Math II · S-CP.9 71 of 73
Use permutations and combinations to compute probabilities of compound events and solve problems.

Count permutations when order matters

Problem
For five distinct books, multiply the available positions 5×4×3×2×1 and express the permutation count using factorial notation.
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Hint

Notice that all 5 books are distinct and all 5 are being arranged, so use a factorial.

When arranging \(n\) distinct objects in order, the number of permutations is \(n!\).

Solution walkthrough

01

Decide whether order matters

\[\text{five distinct books arranged in a row}\]

Changing which book occupies any position changes the arrangement, so the outcomes are ordered permutations.

02

Count choices by position

\[5\cdot4\cdot3\cdot2\cdot1\]

There are 5 choices for the first position, then 4, 3, 2, and 1 choices as books are used.

03

Express the product as a factorial

\[5\cdot4\cdot3\cdot2\cdot1=5!\]

By definition, 5 factorial is the descending product of the positive integers from 5 through 1.

04

Evaluate and interpret

\[5\ne~120\]

The multiplication gives 120, so five distinct books can be placed in a row in 120 different orders.

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Another way

  1. Use the permutation formula 5P5=5!/(5-5)!=5!/0!=120.

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Common mistake

Do not use a combination: two orders containing the same books are different arrangements when positions matter.