Binomial Expansion

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$(x+y)^n = \sum_{k=0}^n {n \choose k}x ^k y^{n-k}$

Using this generalized binomial form, we can determine a particular term

of the expansion. In a common problem, you're asked to find the constant

term of the expansion.

Problem Find the constant term in the binomial expansion of $(3x^2 + \frac{2}{x^3<br />})^{15}$ .

We have $n=15$ and let $x=3x^2$ and $y = \frac{2}{x^3}$ . We must find $k$

such that $x^k y^{n-k}$ eliminates the variable. That is, $x^{2k} \cdot</p> <p>x^{-3(15-k)} = 1 = x^0$ or $2k-3(15-k)=0$ , so $k = 9$ . We then find the

constant term as usual.

We can further generalize to polynomial expansion.

Problem What is the coefficient of $x^5y^2z^2$ in the expansion of $(x+y+2z)^9$ ?

Problems used in this document:
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