A math professor has 4 freshmen 5 sophomores 6 juniors and 3

A math professor has 4 freshmen, 5 sophomores, 6 juniors and 3 seniors in his class. He wants to choose a team of 5 students for a competition. How many different teams are possible? How many different teams would include all 3 seniors? How many teams consisting of 2 sophomores and 3 juniors are possible? How many teams are possible where each rank is represented?

Solution

a.

Total:4+5+6+3=18 studnets

So number of different teams is:C(18,5)

b.

All seniors are chosen in 1 ways

Remaining 2 members ar chosen from 13 studnetsin C(13,2)

So, C(13,2) teams

c.

2 sophomores are selected in C(5,2)

3 juniors are selected in C(6,3)

So, C(5,2)C(6,3) teams

d.

So one student is selected from each rank in

C(4,1)C(5,1)C(6,1)C(3,1)=360 ways

So we have to only select 1 more student from 18-4 =14 students

This is done in C(14,1)=14 ways

So, 14*360 teams