Question

From $6$ different novels and$3$ different dictionaries, $4$ novels and$1$ dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is

• A. at least$500$ but less than$750$
• B. at least$750$ but less than$1000$
• C. at least$1000$
• D. less than$500$

Answer 1

The correct option is C

at least$1000$

Explanation for correct option:

Step 1: Select $\mathbf{4}$ novels and$\mathbf{1}$ dictionary from $\mathbf{6}$ different novels and $\mathbf{3}$ different dictionaries.

We have$6$ novels and$3$ dictionaries. We can select$4$ novels and$1$ dictionary in

$$\begin{gathered} {}^{6}C_4\times {}^{3}C_1=\frac{6!}{2!4!}\times \frac{3!}{2!1!} \\ \Rightarrow \frac{6\times 5\times 3}{2}=45 \end{gathered}$$

We can select$4$ novels and$1$ dictionary in$45$ ways.

Step 2: $\mathbf{4}$ novels and $\mathbf{1}$ dictionary are to be arranged.

Now, $4$novels and $1$ dictionary are to be arranged so that dictionary is always in middle. So remaining$4$ novels can be arranged in$4!$ ways.

Hence total arrangements possible are

$45\times 24=1080$ ways.

Hence, the correct option is $\mathbf{\left(}\mathit{C}\mathbf{\right)}$ at least$\mathbf{1000}$