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 $1000$

Number of novels $=6$

Also, the number of dictionaries $=3$

Number of ways of selecting $4$ novels and $1$ dictionary is ${}^6{C}_4{\times }^3{C}_1$

Arrangements such that the dictionary is always in the middle is given by :

$NNDNN$

The position of the dictionary is fixed. The novel can be arranged in $4!$ different ways

Total number of ways we can arrange $4$ novels and $1$ dictionary so that dictionary always in the middle

${=}^6{C}_4\times 4!{\times }^3{C}_1$

$=15\times 24\times 3=1080$

Option C is correct.