Question

# In a group of 2 freshmen, 3 Sophomores, and 10 Junior in how many ways 5...

In a group of 2 freshmen, 3 Sophomores, and 10 Junior in how many
ways 5 students be chosen if:
a. regardless of their year level?
b. the two freshmen must be included?
c. the two freshmen must be excluded?
d. there must be at least 4 juniors?
e.there must be at most 3 sophomores?

Total = 2 + 3 + 10 = 15 students.

a) Total ways = C(15, 5) =

Hence, total 3003 ways are there.

b) 2 freshman must be included.

So, we need (2 out of 2 freshman) and (3 out of 13 other students)

Total ways = C(2, 2) * C(13, 3) =

Hence, total 286 ways are there.

c) 2 freshman must be excluded.

So, we need 5 out of 13 other students.

Total ways = C(13, 5)=

Hence, total 1287 ways are there.

d) At most 3 Sophomores.

This is the case of all cases because only 3 Sophomores are there and we can take 0, 1, 2 or 3 Sophomores which are only possible combination here.

Hence, total ways = 3003 (from part a)

I hope these will help. Thank you.