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Math Help - Combinations?

  1. #1
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    Combinations?

    Not sure if this is the right area for combinations/pascal triangle questions...


    A)What is a constant in Binomial expansion? Example, I have to state whether or not a constant appears in the expansion of (x+y)^11, and explain why. A quick explanation, pls?

    B) Sandra is testing to see if she can tell the difference between diet pop, and regular pop. She knows there are 3 cups of diet pop, and 2 cups of regular pop, and she'll write the order which she believes the pop to be in. How many ways can record D(for Diet) and R(for Regular) if order matters?

    C)There are 8 non-fiction books, and 7 fiction books. You are to recieve two of them as a gift, how many combinations are possible if you are to recieve AT LEAST 1 fiction book?

    -Is this as simple just figuring out how many combinations of 2 fiction books there are, and multiplying it by the combinations of 1 fiction book, 1 non-fiction book(which would be 56 right?)?

    D) How do you solve for a constant tern in the expansion of (x+2)^9?


    Mucho Gracias to any and all help!
    Last edited by Blahdkm; May 25th 2009 at 03:13 AM.
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  2. #2
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    Combinatorics

    Hello Blahdkm
    Quote Originally Posted by Blahdkm View Post
    Not sure if this is the right area for combinations/pascal triangle questions...


    A)What is a constant in Binomial expansion? Example, I have to state whether or not a constant appears in the expansion of (x+y)^11, and explain why. A quick explanation, pls?
    A constant term is one that doesn't involve any variables - like x and y, for instance. In the example you quote, there are no constant terms - every term involves x or y or both. But in an expression like

    \Big(x + \frac{2}{x^2}\Big)^6

    there will a term that doesn't involve x. You find it as follows:

    \Big(x + \frac{2}{x^2}\Big)^6 = \Big(\frac{1}{x^2}\Big)^6 ( x^3 + 2)^6

    Since the term outside the bracket is \frac{1}{x^{12}}, the term in x^{12} in the expansion of (x^3+2)^6 will result in the cancellation of the x's, and hence a constant term. This term is

    \binom64(x^3)^4.2^2 = 60x^{12}

    So the constant term in the expansion of \Big(x + \frac{2}{x^2}\Big)^6 is 60.


    B) Sandra is testing to see if she can tell the difference between diet pop, and regular pop. She knows there are 3 cups of diet pop, and 2 cups of regular pop, and she'll write the order which she believes the pop to be in. How many ways can record D(for Diet) and R(for Regular) if order matters?
    The D's can be placed in 3 positions out of 5, and the R's in the remaining positions. So the number of possible arrangements is \binom53 = 10.

    C)There are 8 non-fiction books, and 7 fiction books. You are to recieve two of them as a gift, how many combinations are possible if you are to recieve AT LEAST 1 fiction book?

    -Is this as simple just figuring out how many combinations of 2 fiction books there are, and multiplying it by the combinations of 1 fiction book, 1 non-fiction book(which would be 56 right?)?
    This is only part of the answer - where you have exactly one fiction book. You need to add to this the number of ways of choosing 2 fiction books from 7, which is \binom72 = 21. So the total number is 56+21=77.

    D) How do you solve for a constant tern in the expansion of (x+2)^9?


    Mucho Gracias to any and all help!
    Answer 2^9 = 512. See A above!

    Grandad
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  3. #3
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    Many Many thanks friend.


    Last Question, I have a 3d square connected at one point. It has a height of 3 squares, a width of 1 square, and length of 3 squares.

    I'm supposed to find the amount of paths from S(Upper left corner), to U(bottom right corner), with no backtracking. The fact that it's a 3d object is seriously throwing me for a loop.

    How do you determine the amount of paths on a 3d object(I'm assuming you have to use pascals triangle in some way)?
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  4. #4
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    Don't add a new question

    Hello Blahdkm
    Quote Originally Posted by Blahdkm View Post
    Many Many thanks friend.


    Last Question, I have a 3d square connected at one point. It has a height of 3 squares, a width of 1 square, and length of 3 squares.

    I'm supposed to find the amount of paths from S(Upper left corner), to U(bottom right corner), with no backtracking. The fact that it's a 3d object is seriously throwing me for a loop.

    How do you determine the amount of paths on a 3d object(I'm assuming you have to use pascals triangle in some way)?
    Post this as a new thread - you're more likely to get a response that way. And you'll be sticking to the rules as well! (See Rule 14)

    Thanks.

    Grandad
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