Question

How many ways can $6$ different books be arranged on a shelf?

Answer 1

Calculate the number of ways that $6$ different books be arranged on a shelf.

The given number of books $\left(n\right)$ is $6$.

The permutations should be used to calculate the number of ways $n$ elements can be arranged in a sequence:

$\begin{array}{rcl}{}_{}{}^{n}P_r& =& {}_{}{}^{6}P_6[\because r=6]\\ & =& \frac{6!}{(6-6)!}\left[\because {}_{}{}^n{P}_r=\frac{n!}{(n-r)!}\right]\\ & =& 6!\left[\because 0!=1\right]\end{array}$

Multiply all natural numbers between $1$ and $6$ to determine the factorial:

$\begin{array}{rcl}6!& =& 1\times 2\times 3\times 4\times 5\times 6\\ & =& 36\times 20\\ & =& 720\end{array}$

Therefore, the $6$ different books can be arranged in $720$ ways on a shelf.