Three officers a president a treasurer and a secretary re t

Three officers - a president, a treasurer and a secretary >>re to be chosen from among 6 people: A, B, C, D, E, F. Suppose that A, E cannot he treasurer, A, D cannot be a president, and D, C must he a secretary. How many ways can the officers be chosen?

Solution

In this problem, the conditions \'secretary has to be C and D and D cannot be president\' leads to two separate cases and the final answer is the sum of answers to these two cases.

Case 1

C is the secretary.

Then, since president cannot be A or D and C is already selected, president can only be any one of B, E and F and hence there are ³C₁ = 3 ways of selecting the prsident.

For each one of these 3 selections of president, the treasurer can be any one of B, D and F, but not B or F depending on who was selected as president in the earlier step. Thus, there are only two options, and hence there are ²C₁ = 2 ways of selecting the treasurer.

So, for case 1, number of selections = 1 x 3 x 2 = 6 ......... (1)   

Case 2

D is the secretary.

Then, since president cannot be A or D of whom D is already selected as secretary, president can be any one of B, C, E and F and hence there are ⁴C₁ = 4 ways of selecting the prsident.

For each one of these 4 selections of president, the treasurer can be any one of B, C, D and F, but not B, C or F depending on who was selected as president in the earlier step. Thus, there are only three options, and hence there are ³C₁ = 3 ways of selecting the treasurer.

So, for case 2, number of selections = 1 x 4 x 3 = 12 ......... (2)

Combining (1) and (2), total number of selctions = 6 + 12 = 18 (ANSWER)