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.

• 70
• 140
• 210
• 420
• 840

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: B

We seek the number of ways to select 4 individuals from 7 and choose one selected member as subcommittee chair. (The existence of a subcommittee secretary is irrelevant.) There are (7 choose 4) = 7(6)(5)(3)/4! = 35 ways to form a collection of 4 individuals from 7. For each of them, there are 4 ways to assign a chair. The product, 140, is the number of different ways to form a subcommittee of 4 individuals and assign a chair and thus is the maximum number without repetition.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.