Chapter 12.5, Problem 50AYU

To determine

To calculate:The value of $12!,20!\text{and}\text{25!}$

Explanation of Solution

The expression for factorial is calculated as,

  $n!\approx \sqrt{2n\pi }{\left(\frac{n}{e}\right)}^n\left(1+\frac{1}{12n-1}\right)$

For $n=12$

  $\begin{array}{c}12!\approx \sqrt{2\left(12\right)\pi }{\left(\frac{12}{e}\right)}^{12}\left(1+\frac{1}{12\left(12\right)-1}\right)\\ \approx 479013972.4\end{array}$

For $n=20$

  $\begin{array}{c}20!\approx \sqrt{2\left(20\right)\pi }{\left(\frac{20}{e}\right)}^{20}\left(1+\frac{1}{12\left(20\right)-1}\right)\\ \approx 2.432924\times {10}^{18}\end{array}$

For $n=25$

  $\begin{array}{c}25!\approx \sqrt{2\left(25\right)\pi }{\left(\frac{25}{e}\right)}^{25}\left(1+\frac{1}{12\left(25\right)-1}\right)\\ \approx 1.5511299\times {10}^{25}\end{array}$