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Thread: 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 $\displaystyle (x+y)^n=\sum_{k=0}^{n}{n\choose k}x^{n-k}y^k$. 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 $\displaystyle n+1$...............so
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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 $\displaystyle (a+b)^1$ have?

    What about $\displaystyle (a+b)^2$?

    $\displaystyle (a+b)^3$?

    :
    :

    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

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

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

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