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 she. If 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 $1000$

Out of $6$ novels, $4$ novels can be selected in ${}^6{C}_4$ ways.
Also out of $3$ dictionaries, $1$ dictionary can be selected ${}^3{C}_1$ ways.
Since, the dictionary is fixed in the middle, we only have to arrange $4$ novels which can be done in $4!$ ways.
Then the number of ways ${}^6{C}_4{\cdot }^3{C}_1\cdot 4!$ $=\frac{6.5}{2}\cdot 3.24=1080$