Results 1 to 3 of 3

Thread: Finding a Binomial Coefficient

  1. #1
    Junior Member
    Joined
    Oct 2011
    Posts
    28

    Thumbs up Finding a Binomial Coefficient

    What is the coefficient of x^5 in the binomial expression (x - 2x^-2)^20 when multiplied by (1+3x^3)



    So far I have:
    (20 over k)(-1)^k (2)^k x^(20-3k)

    so then 20-3k = 5 such that k = 5

    So then (20 over 5 )(-1)^5 (2)^5

    But then I don't know how to handle multiplying it by ( 1 + 3x^3)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790

    Re: Finding a Binomial Coefficient

    $\displaystyle (x - 2x^{-2})^{20}=a_{20}x^{20}+a_{17}x^{17}+\dots+a_5x^5+a_ 2x^2+\dots+a_{-37}x^{37}+a_{-40}x^{-40}$ for some coefficients $\displaystyle a_k$. Also, $\displaystyle 1+3x^3=b_0x^0+b_3x^3$ for $\displaystyle b_0=1$, $\displaystyle b_3=3$. The coefficient of $\displaystyle x^5$ in $\displaystyle (x - 2x^{-2})^{20}(1+3x^3)$ is $\displaystyle \sum_{i+k=5}a_ib_k$, i.e., $\displaystyle a_2b_3+a_5b_0$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,796
    Thanks
    2827
    Awards
    1

    Re: Finding a Binomial Coefficient

    Quote Originally Posted by ThatPinkSock View Post
    What is the coefficient of x^5 in the binomial expression (x - 2x^-2)^20 when multiplied by (1+3x^3)
    So far I have: (20 over k)(-1)^k (2)^k x^(20-3k)
    so then 20-3k = 5 such that k = 5
    So then (20 over 5 )(-1)^5 (2)^5
    but then I don't know how to handle multiplying it by ( 1 + 3x^3)
    I have a different take on this question from you and reply #2.
    Each term in $\displaystyle \left(x-2x^{-2}\right)^{20} $looks like $\displaystyle \binom{20}{k}(-2)^{20-k}x^{3k-40}$
    Now you want $\displaystyle x^5$ from $\displaystyle [x^{3k-40}]~\&~3x^3{x^{3k-40}}$
    So $\displaystyle k=14~\&~k=15$ WHY?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Sum with binomial coefficient
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Aug 11th 2011, 11:00 AM
  2. Binomial theporem : Finding the Coefficient
    Posted in the Discrete Math Forum
    Replies: 9
    Last Post: Oct 31st 2010, 04:04 PM
  3. Binomial Coefficient
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Nov 1st 2009, 05:37 PM
  4. Binomial Theorem or Binomial Coefficient
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: Oct 2nd 2009, 01:06 PM
  5. Binomial Coefficient
    Posted in the Pre-Calculus Forum
    Replies: 16
    Last Post: Apr 9th 2008, 07:13 PM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum