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Math Help - binomial theorem

  1. #1
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    binomial theorem

    can someone pls help me solve the following expansion?

    a binomial expansion, (a+b)^n, has 15 terms altogether. what is the value of n?

    this question is part of my combinatorics chapter assignment. i don't know where else to put it
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by oryxncrake View Post
    can someone pls help me solve the following expansion?

    a binomial expansion, (a+b)^n, has 15 terms altogether. what is the value of n?

    this question is part of my combinatorics chapter assignment. i don't know where else to put it
    Hello. We know that (x+y)^n=\sum_{k=0}^{n}{n\choose k}x^{n-k}y^k. We know that no two of these terms can be coalesced into a single term (since they all have differing powers of x and y). Thus the number of terms is n+1...............so
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  3. #3
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    where does n+1 come from?
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  4. #4
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    also, does the "15" go anywhere in the equation?
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  5. #5
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    Quote Originally Posted by oryxncrake View Post
    also, does the "15" go anywhere in the equation?
    How many terms does (a+b)^1 have?

    What about (a+b)^2?

    (a+b)^3?

    :
    :

    notice the pattern?

    CB
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Yes. A better way maybe just to notice what CaptainBlack said.

    What I was getting at was that

    \left(x+y\right)^n=\sum_{\ell=0}^{n}{n\choose \ell}x^{n-\ell}y^\ell

    And noting that on the RHS no two terms can be coalesced into a single term we can deduce that \left(x+y\right)^n has as many terms as \sum_{\ell=0}^{n}x^{n-\ell}y^\ell. But how many does that have?

    Well it not only has the first term through the nth term (which is n terms) but it has the 0th term. So the number of terms in \sum_{\ell=0}^{n}x^{n-\ell}y^\ell is n+1. Consequently the number of terms in \left(x+y\right)^n is n+1
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