Problem 1 is:
A class contains 7 freshmen, 7 sophomores, 7 juniors and 7 seniors. A committee of size 10 is formed. Find the number of committees with
a. all seven seniors
I have: C(7,7)*C(21,3)
b. 2 freshmen, 3 sophomores, 1 junior, 4 seniors
I have: C(7,2)*C(7,3)*C(7,1)*C(7,4)
c. 2 each from two classes, 3 from a third class and 4 from the remaining class.
I...am not too sure about this one. I want to say C(4,2)*2*C(7,2)2*C(7,3)*C(7,4). But... like I said I am not sure.
My reasoning is that I have 4 types, I need to choose from 2 of those. Then I choose 2 from seven of each type (hence the first *2). Afterwards, I have 2 options to choose the 3 from. Hence the 2*C(7,3). Then finally I choose 4 from the last group of classmates.
Problem 2 is:
There are 40 books (20 Math, 15 CS, and 5 English) on a shelf, all different.
a. Fine the number of arrangements of the books in which.
1. all books in the same subject area are grouped together.
I treated each subject area as one object. So I have 3! ways to arrange those "three" objects.
2. all CS books are grouped together.
I treated the CS books as one object, so I got 26!
I guess for these problems I am wanting feedback on if I am doing the problems correctly. Thanks.