Combinatorics

imane81 wrote:Hi,

Please if you could help in this one, many thanks!!!

A certain university will select 1 of 7 candidates eligible to fill a position in the mathematics department and 2 of 10 candidates eligible to fill 2 identical positions in the computer science department. If none of the candidates is eligible for a position in both departments, how many different sets of 3 candidates are there to fill the 3 positions?
A. 42
B. 70
C. 140
D. 165
E. 315
Many thanks!!![/spoiler]

We can take the task of filling both positions and break it into 2 stages.
Stage 1: Fill the 1 math position
Stage 2: Fill the 2 CS positions

Stage 1: 7 candidates and 1 position --> 7 ways to accomplish stage 1
Stage 2: 10 candidates and 2 positions. Note that the order in which we select candidates doesn't matter. For example, selecting candidate B and then candidate C is the same as selecting candidate B and then candidate C.
So, we can use combinations. The numbers of ways to accomplish stage 2 is 10C2 (45)

So, the total number of ways to complete stage 1 and stage 2 equals 7 x 45 = 315 (E)