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Math Help - Combination Problem.

  1. #1
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    Combination Problem.

    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.
    Last edited by Alterah; May 4th 2010 at 01:45 AM.
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  2. #2
    MHF Contributor
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    Hello Alterah
    Quote Originally Posted by Alterah View Post
    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)
    Correct!

    b. 2 freshmen, 3 sophomores, 1 junior, 4 seniors
    I have: C(7,2)*C(7,3)*C(7,1)*C(7,4)
    Correct!

    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.
    You're right to be uncertain. I think you've gone a bit adrift here.

    It's easiest if you first choose the classes that will contribute the 3 and the 4 students, because these numbers only occur once. Then choose 2 students from each of the remaining 2 classes. So that's:
    4 \times C(7,3) \times 3 \times C(7,4) \times C(7,2)\times C(7,2)
    OK?

    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.
    OK so far, but you've then got to arrange the books within each 'object'. So you'll need to multiply by 20!\times15!\times5!.

    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.
    Again, you have then got to arrange the CS books within their 'object'. So you'll multiply by 15!.

    Grandad
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  3. #3
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    Thanks. For the second problem I realized that after the fact. So, I have adjusted accordingly. And thanks for your feedback on part c. Makes sense to do it that way.
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