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Find the coefficient of $x^3 y^4$ in the expansion of $(2x-4y)^7$. I would also like an explanation for how the final answer was obtained.

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  • $\begingroup$ Binomial expansion \begin{eqnarray*} (2x-4y)^7= \cdots +\binom{7}{3}(2x)^3 (-4y)^4 +\cdots \end{eqnarray*} $\endgroup$ – Donald Splutterwit Sep 11 '18 at 1:26
  • $\begingroup$ How did you choose that combination and what would the final answer be? $\endgroup$ – Murph Jones Sep 11 '18 at 1:29
  • $\begingroup$ \begin{eqnarray*} (a+b)^n= \cdots +\binom{n}{r}a^r b^{n-r} +\cdots \end{eqnarray*} What term do you need to choose to obtain the monomial $x^3 y^4$ ? ... simplify ? You tell me what you think. $\ddot \smile$ $\endgroup$ – Donald Splutterwit Sep 11 '18 at 1:34
  • $\begingroup$ n=7 and r would have to equal 3. I just don't understand how this summation (if it is a summation) works $\endgroup$ – Murph Jones Sep 11 '18 at 1:45
  • $\begingroup$ I do understand how you assigned values to r and n the question now is just how the recipe works $\endgroup$ – Murph Jones Sep 11 '18 at 1:46
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The Binomial Theorem states $$(a+b)^n= \sum_{r=0}^{n}\binom{n}{ r} (a)^{n-r}(b)^r$$ and so $$(2x-4y)^7=\sum_{r=0}^{7}\binom{7}{ r} (2x)^{7-r}(-4y)^r.$$

So, we simply need to consider the $5$th term (i.e., when $r=4$): $$\binom{7}{ 4} (2x)^{7-4}(-4y)^4=\binom{7}{4}2^34^4x^3y^4.$$

I’ll leave it to you to evaluate what this equals.

In general, whenever you are asked to find a particular coefficient in the Binomial expansion, try to reword the question into one which asks “What value of $r$ will give me the desired term?” and then evaluate $$\binom{n}{r}(a)^{n-r}(b)^{r}$$ for that particular $r$, as I did above.

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It is convenient to use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$.

We obtain \begin{align*} \color{blue}{[x^3y^4]}&\color{blue}{(2x-4y)^7}\\ &=[x^3y^4]\sum_{k=0}^7\binom{7}{k}(2x)^k(-4y)^{7-k}\tag{1}\\ &=[y^4]\binom{7}{3}2^3(-4y)^4\tag{2}\\ &=\binom{7}{3}2^3(-4)^4\tag{3}\\ &\,\,\color{blue}{=71\,680} \end{align*}

Comment:

  • In (1) we apply the binomial theorem.

  • In (2) we select the coefficient of $x^3$.

  • In (3) we select the coefficient of $x^4$.

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