A chair, a vice-chair and a secretary are chosen, one each, from 5 people. If no one may hold two or more roles, which of the following is the number of possible choices?

Answer: B. 60

Because the roles differ, the number of permutations 5P3 is needed, which also distinguishes who is assigned to which role. There are 5 choices for the chair, 4 of the remaining people for the vice-chair and 3 of those still remaining for the secretary, so 5 × 4 × 3 = 60. Multiplying the number of combinations without roles, 5C3 = 10, by the 3! = 6 ways of assigning the roles to the three people also gives 60.

• A. This is the number of combinations 5C3, which ignores the roles. Even for the same three people, there are 6 different assignments of who takes which role.
• B. Correct. 5P3 = 5 × 4 × 3 = 60.
• C. This is 5³ = 125, which allows one person to hold several roles.