EF Factorials

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Section 21.1 Factorials

In counting, factorials come up a lot.

$n!$: 🔗

🔗
for natural number $n,$ notation for the computation formula

$n (n - 1) (n - 2) \dots 2 \cdot 1$

Example 21.1.1. Two factorial calculations.

\begin{aligned} 3! & = 3 \cdot 2 \cdot 1 = 6, & 7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5, 040. \end{aligned}

Example 21.1.2. Factorial factors.

A factorial contains every smaller factorial as a factor. For example,

\frac{7!}{3!} = \frac{7 \cdot 6 \cdot 5 \cdot 4 \cdot (3!)}{3!} = 7 \cdot 6 \cdot 5 \cdot 4 = 840 .

Convention 21.1.3.

To avoid division by zero in certain formulas, define

0! = 1 .

This choice is also made to be consistent with the methods for counting permutations we will explore in this chapter.

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