Chapter 11.8, Problem 27WE

To determine

True or False:

  $\left(2n\right)!=2^n\left(n!\right)$

Answer to Problem 27WE

The statement is false.

Explanation of Solution

Given information:

Statement:

  $\left(2n\right)!=2^n\left(n!\right)$

Consider $\left(2n\right)!=2^n\left(n!\right)$

Put, $n=2$

  $\begin{array}{l}\left(2n\right)!=2^n\left(n!\right)\\ \left(2\left(2\right)\right)!=2^2\left(2!\right)\\ \left(4\right)!=4\left(2\right)\\ 24=8\\ 24\ne 8\end{array}$

Hence the statement is false.