In how many ways can you seat 7 people on a bench if one of

Solution:

If there is no restriction on the seating arrangement, there will be 7! = 5040 ways to seat the 7 people.

If Rohit must sit on the middle seat, there will be 6! = 720 arrangements. However, since he can’t sit there, we must subtract 720 from 5040. Likewise, since he can’t sit at either end, we have to further subtract 720 twice (once for each end) from 5040. In other words, we have to subtract 3 times 720 from 5040. Therefore, there are 5040 - 3(720) = 5040 - 2160 = 2880 seating arrangements if Rohit does not want to sit on the middle seat or at either end.

Alternate Solution:

For the first seat, there are 6 options since Rohit will not sit on the first seat. For the fourth seat (which is the middle seat), there are 5 options since we must exclude Rohit and the person chosen for the first seat. Likewise, there are 4 options for the last seat since we are excluding Rohit and the two previously selected people.

For the second seat, there are also 4 options since we now include Rohit. For the third, fifth and sixth seats, there are 3, 2, and 1 available options, respectively. Thus, the number of arrangements for the 7 people to sit as described in the question is 6 x 5 x 4 x 4 x 3 x 2 x 1 = 2,880.

Answer: C