#### A group of 8 people, including 3 men and 5 women, is to be seated in a row of 8 chairs. In how many different ways can the 3 men be seated together?Option: 1 24000Option: 2 36000Option: 3 49000Option: 4 81000

Given that,

There are 8 people who must be seated in a row on 8 chairs.

Therefore, if we find the number of ways in which all 3 men occupy consecutive seats and subtract this number from the total number of ways in which the 8 people can be arranged among themselves, we will get the required answer.

The 8 people can be arranged among themselves in ${}^8 P _5$ ways

Thus,

${}^8 P _5=40320$

Assume that the 3 men are one entity. The total number of ways in which they can be arranged among themselves is 3! Ways.

Also, the set of 3 men and the other people can be arranged among themselves in 6! ways.

Thus, the total number of ways in which 3 men are together is given by,

$3! \times 6!=6 \times 720\\ 3! \times 6!=4320$

Thus, the number of ways in which all 3 men will not occupy consecutive seats is given by,

$40320-4320=36000$

Therefore, the total number of ways to arrange the people is 36000.