Question

"Each week, a subcommittee of four individuals is formed from among the members of a committee comprising seven individuals. Two subcommittee members are then assigned to lead the subcommittee, one as chair and the other as secretary.Calculate the maximum number of consecutive weeks that can elapse without having the subcommittee contain four individuals who have previously served together with the same subcommittee chair."

Answer 1

Answer:

140 weeks.

Step-by-step explanation:

We are being asked for the maximum number that can elapse without having 2 subcommitees with the same members and the same chair. Notice that the secretary is not mentioned in this statement, so we can ignore that position.

Now, the maximum number of weeks would be equal to the total possible combinations of subcommitees that can be formed from the 7 individuals commiteed, times the number of options that can sit at the chair possition.

The total possible combinations of subcommitees that can be formed from the 7 individuals can be expressed like the following expression. Notice that there is no a particular order to be considered, so we use the combination formula:

$\left[\begin{array}{ccc}7\\4\end{array}\right]$ = $\frac{7!}{4!(7-4)!}$ = 35

Now, each of the 4 individuals have the same probability of being chosen as the chair commitee. So, we have to multiply 4 times the number of possible subcommitees in order to arrive our answer:

x = 35 * 4 = 140

This would be the maximum number of subcommitees that can be formed with no subcommitees with the same members and the same chair occurring.