Question

From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then, 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 $1000$.

Explantion for the correct option:

Step: 1- Find out the ways of selecting $4$ novels from $6$ different novels.

The different ways $4$ novels can be selected from $6$ different novels is:

$$\begin{gathered} ={}^6{C}_4 \\ =\frac{6!}{4!\times 2!} \\ =\frac{6\times 5\times 4!}{2\times 4!} \\ =15 \end{gathered}$$

Step: 2- Find out the ways of selecting $1$ dictionary from $3$ different dictionaries.

The different ways $1$ dictionary can be selected from $3$ different dictionaries is:

$$\begin{gathered} ={}^3{C}_1 \\ =3 \end{gathered}$$

Step: 3- Find out the number of ways so that the dictionary get placed in the middle among the selected novels always.

Number of ways of selecting $4$ novels and $1$ dictionary and arrange them in such way that dictinary alwasy come in middle.

$$\begin{gathered} =15\times 3\times 4! \\ =1080 \end{gathered}$$

Hence, option ‘C’ is correct.