Question

In a three-storey building, there are four rooms on the ground floor, two on the first and two on the second floor. If the rooms are to be alloted to six persons, one person occupying one room only, then number of ways in which this can be done so that no floor remains empty is

• A. ${}^8{P}_6-2(6!)$
• B. ${}^8{P}_6$
• C. ${}^8{P}_5(6!)$
• D. ${}^8{P}_6-(6!)$

Answer 1

The correct option is A ${}^8{P}_6-2(6!)$

The number of ways of allotment without any restriction is ${}^8{P}_6$

Now, it is possible that all rooms of second floor or third floor are not occupied. Thus, there are two ways in which one floor remains unoccupied.

Hence, the number of ways of allotment in which a floor is unoccupied is $2\times {}^6{P}_6$

Therefore, the number of ways in which none of the floor remains unoccupied $={}^8{P}_6-2{}^6{P}_6={}^8{P}_6-2(6!)$