Chapter 8, Problem 1RE

For Exercises 1-2, simplify the expression.

$\frac{8!}{3!\cdot 5!}$

To determine

To calculate: The value of the expression $\frac{8!}{3!\cdot 5!}$.

Answer to Problem 1RE

Solution:

The required value is $56$.

Explanation of Solution

Given Information:

The provided expression is:

$\frac{8!}{3!\cdot 5!}$

Formula used:

Let $n$ be a positive integer. The expression $n!$ is defined as:

$n!=\left(n\right)\left(n-1\right)\left(n-2\right)\cdots \left(2\right)\left(1\right)$

Calculation:

Consider the provided expression:

$\frac{8!}{3!\cdot 5!}$

Use the mentioned formula to calculate the value of the provided expression:

$n!=\left(n\right)\left(n-1\right)\left(n-2\right)\cdots \left(2\right)\left(1\right)$

Therefore,

$\begin{array}{c}\frac{8!}{3!\cdot 5!}=\frac{\left(8\right)\cdot \left(8-1\right)\cdot \left(8-2\right)\cdot \left(8-3\right)!}{\left(3\right)\cdot \left(3-1\right)\cdot \left(3-2\right)\cdot 5!}\\ =\frac{8\cdot 7\cdot \cancel{6}\cdot \cancel{5!}}{\cancel{3}\cdot \cancel{2}\cdot 1\cdot \cancel{5!}}\\ =8\cdot 7\\ =56\end{array}$

Hence, the simplified expression for $\frac{8!}{3!\cdot 5!}$ is $56$.