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Chapter 11: Problem 14

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{4} C_{4}\)

Short Answer

Expert verified
The evaluation of \({ }_{4} C_{4}\) is 1.

Step by step solution

01

Clarify the Problem

We are asked to compute \({ }_{4} C_{4}\), which means we need to find the number of ways 4 items can be chosen from a group of 4 items.
02

Implement the Combination Formula

We use the combination formula \[ { }_{n} C_{r} = \frac{n!}{r!(n - r)!} \] Here, n = 4 and r = 4. Substituting these values into the formula, we find \[ { }_{4} C_{4} = \frac{4!}{4!(4 - 4)!} \].
03

Simplify the Expression

The factorial of 4, written as 4!, equals 4*3*2*1 = 24. And for any number, factorial of 0, 0!, is defined to be 1. So, \[ { }_{4} C_{4} = \frac{24}{24 * 1} \]. When you simplify this, the 24 in the numerator and the 24 in the denominator cancel out, leaving \({ }_{4} C_{4} = 1\). This means that there is 1 way to choose 4 items from a group of 4 items.

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