Question

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?

Answer #1

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.

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