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Coefficient of $x^3$ in $(1+x^2)(1+x)^{100}$

Coefficient of $x^{10}$ in $(1+x)^{10}(1-x)^{10}$

Coefficient of $x^n$ in $\dfrac{2+x}{2-x}$.

Any help on these would be appreciated.

  • 1
    For the second one, note that we are looking at $(1-x^2)^{10}$ and use the known expansion of $(1+w)^{10}$. For the third one, rewrite as $\frac{1+x/2}{1-x/2}$ and use the known expansion of $\frac{1}{1-w}$.2012-03-22

2 Answers 2

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  1. There are two ways to obtain $x^3$ in $(1+x^2)(1+x)^{100}$: when you multiply $1$ on the first binomial by the $x^3$-term in the expansion of $(1+x)^{100}$, and when you multiply $x^2$ by the $x$-term in the expansion of $(1+x)^{100}$.

  2. You obtain $x^{10}$ by multiplying the $x^i$ term in the expansion of $(1+x)^{10}$ with the $x^{10-i}$ term in the expansion of $(1-x)^{10}$, for $i=0,1,\ldots,10$.

  3. Expand $(2-x)^{-1}$ and figure out the $x^n$ and $x^{n-1}$ terms.

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  1. $(1+x^2)\left({100 \choose 1}x+\cdots+{100 \choose 3}x^3\right) \Rightarrow $ coefficient of $x^3 = {100 \choose 1}+{100 \choose 3} = 100\left(1+33 \times 49\right)= 100 \times 1618= 161800$

Follow suggestion given by Arturo. The one shown above is just to clear things for you.

Do similarly approach the other answers