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Thread: Discrete Math question

  1. #1
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    Question Discrete Math question


    Whats the easiest way to approach questions like this?

    Part (c) in particular.

    Is the first answer 10c4 * 3^4 and last one 15c4 * 4c4?

    Please can someone confirm?
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  2. #2
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    Re: Discrete Math question

    use the binomial theorem

    $(a + b)^n = \sum \limits_{k=0}^n~\dbinom{n}{k}~a^k~b^{n-k}$

    $(3x+y)^{10} = \sum \limits_{k=0}^{10}~\dbinom{10}{k}~(3x)^k~y^{10-k}$

    we are clearly looking for the term that corresponds to $k=4$ which is

    $\dbinom{10}{4}~3^4 x^4 y^6$

    and thus the coefficient is

    $\dbinom{10}{4}~3^4 = 17010$

    use the same method on (b) and (c)
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  3. #3
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    Re: Discrete Math question

    Quote Originally Posted by romsek View Post
    use the binomial theorem

    $(a + b)^n = \sum \limits_{k=0}^n~\dbinom{n}{k}~a^k~b^{n-k}$

    $(3x+y)^{10} = \sum \limits_{k=0}^{10}~\dbinom{10}{k}~(3x)^k~y^{10-k}$

    we are clearly looking for the term that corresponds to $k=4$ which is

    $\dbinom{10}{4}~3^4 x^4 y^6$

    and thus the coefficient is

    $\dbinom{10}{4}~3^4 = 17010$

    use the same method on (b) and (c)
    Part (c) has 3 variables
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  4. #4
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    Re: Discrete Math question

    Quote Originally Posted by inayat View Post
    Part (c) has 3 variables
    apply twice, adapt, improvise, overcome!

    $\begin{align*}

    &(x+y+3)^{15} = ((x+y)+3)^{15} = \\ \\

    &\sum \limits_{k=0}^{15}~(x+y)^k~3^{15-k} = \\ \\

    &\sum \limits_{k=0}^{15}~\left(\sum \limits_{j=0}^k~x^j~y^{k-j}\right)~3^{15-k}

    \end{align*}$
    Thanks from inayat and topsquark
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  5. #5
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    Re: Discrete Math question

    Quote Originally Posted by inayat View Post
    Part (c) has 3 variables
    Quote Originally Posted by inayat View Post

    Whats the easiest way to approach questions like this?
    Part (c) in particular.
    last one 15c4 * 4c4? NO
    In the expansion of ${(x + y + 3)^{15}}$
    we have ${\dbinom{15}{5}}{(x + y)^{10}}{(3)^5}$
    $ {\dbinom{15}{5}} \cdot {\dbinom{10}{4}}{y^{10 - 4}}{x^4}{(3)^5}$

    You should get $153243090$ see HERE
    Thanks from inayat
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