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Chapter 15: Problem 10

Evaluate \(0 !\)

Short Answer

Expert verified
By convention, 0! is assigned the value 1 to satisfy the unique identity of the mathematical function and simplify formulas involving factorials. Therefore, the value of 0! is equal to 1.

Step by step solution

01

Understand the factorial definition.

A factorial is a function that multiplies a given non-negative integer (n) by all smaller positive integers. In other words, n! = n * (n-1) * (n-2) * ... * 3 * 2 * 1. For example, 4! can be calculated as 4 * 3 * 2 * 1 = 24.
02

Factorial of 0 and unique identity

The value of 0! seems ambiguous according to the factorial definition since there are no positive integers less than or equal to 0. However, by convention, 0! is assigned the value of 1 to satisfy the unique identity of the mathematical function. One reason for this convention is that it helps simplify formulas dealing with factorials. Consider the formula for combinations: \[C(n, r) = \frac{n!}{(n-r)! r!}\] If r = 0 or r = n, then we have either \((n-r)! = 0!\) or \(r! = 0!\). With the convention 0! = 1, the formula simplifies neatly to \(C(n, 0) = C(n, n) = 1\), which states that there is only one way to choose 0 or n items from a set of n items, which is intuitive.
03

Conclude our evaluation of 0!

According to the convention, 0! is assigned a value of 1 to simplify mathematical formulas and maintain consistency in the factorial function. Therefore, the value of 0! is equal to 1.

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