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Chapter 11: Problem 30

Evaluate each factorial expression. \(\frac{8 !}{(8-5) !}\)

Short Answer

Expert verified
The evaluation of the expression \(\frac{8 !}{(8-5) !}\) is 336.

Step by step solution

01

Identify the factorial expressions

The task has provided the expression \(\frac{8 !}{(8-5) !}\). Here, the factorial expressions are \(8 !\) in the numerator and \((8-5) !\) in the denominator.
02

Simplify the factorial in the denominator

The expression in the denominator, \((8-5) !\), simplifies to \(3 !\). The expression becomes \(\frac{8 !}{3 !}\).
03

Break down the factorials

Factorial expressions can be broken down further. The expression \(8 !\) equates to \(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\). The expression \(3 !\) equates to \(3 \times 2 \times 1\). The task has now become \(\frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1}\).
04

Simplify by cancelling out

The numbers \(3, 2, 1\) in the numerator are the same in the denominator, so these can be cancelled out. The expression simplifies to \(8 \times 7 \times 6\).
05

Evaluate the result

By multiplying the expression \(8 \times 7 \times 6\), the final result is 336.

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