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. Then the number of such arrangements is

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

Answer 1

The correct option is A At least $1000$

Number of novels $=6$
Number of dictionaries $=3$
$\therefore$ Number of ways of selecting $4$ novel from $6={}^6{C}_4$ ways.
Number of ways of selecting $1$ dictionary from $3={}^3{C}_1$ ways
When dictionary is always in middle, So $4$ Novel can be arrange in $4!$ ways
$\therefore$ The required number of ways
$={}^6{C}_4\times {}^3{C}_1\times 4!$
$=\frac{6\times 5}{2\times 1}\times 3\times 24$
$=1080$