Question

$n$ books are arranged on a shelf so that two particular books are not next to each other, There were $480$ arrangements altogether. Then the number of books on the shelf is?

• A. $5$
• B. $6$
• C. $10$
• D. $8$

Answer 1

The correct option is B

$6$

Explanation for the correct option:

Finding the number of books on the shelf:

We know that the total number of arrangements of $n$ objects is $n!$.

So, the total number of arrangements of $n$ books on a shelf is $n!$.

First, we find the number of arrangements when two particular books are together.

When two particular books are together, we can assume them as a one book and in total there will be $n-1$ books.

Now, the total number of arrangements of $n-1$ books on a shelf is $(n-1)!$.

Again, those two particular books can be arranged in $2!=2$ ways.

The number of times, two particular books are together, is given by = $2\times (n-1)!$.

Thus, the number of times, when two books are not together is = $n!-2(n-1)!$

Now, the total number of arrangements is $480$. So, we must get :

$$\begin{gathered} n!-2(n-1)!=480 \\ \Rightarrow \left(n-1\right)!\left(n-2\right)=480 \\ \Rightarrow (n-2)(n-1)!=(6-2)(6-1)! \end{gathered}$$

So, by comparing both sides, we get $n=6$

Hence, option (B) is the correct answer.