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Thread: 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
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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:
    $\displaystyle 4 \times C(7,3) \times 3 \times C(7,4) \times C(7,2)\times C(7,2)$
    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 $\displaystyle 20!\times15!\times5!$.

    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 $\displaystyle 15!$.

    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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