1. ## Binomial theorem

Using the binomial theorem, write down expressions for $\displaystyle (1+x)^n , (1-x)^n , (1-{x^2})^n$. By comparing terms in $\displaystyle x^{2r}$, prove that for any r with 0≤r≤n

$\displaystyle {\left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)} = \sum\limits_{k = 0}^{2r}(-1)^{r-k} {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right) } {\left({\begin{array}{*{20}c} n \\ 2r - k \\ \end{array} } \right) }$

Got expansions for the three binomial expansions, but I don't know how to compare the terms in $\displaystyle x^{2r}$

2. Originally Posted by smwatson
Using the binomial theorem, write down expressions for $\displaystyle (1+x)^n , (1-x)^n , (1-{x^2})^n$. By comparing terms in $\displaystyle x^{2r}$, prove that for any r with 0≤r≤n

$\displaystyle {\left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)} = \sum\limits_{k = 0}^{2r}(-1)^{r-k} {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right) } {\left({\begin{array}{*{20}c} n \\ 2r - k \\ \end{array} } \right) }$

Got expansions for the three binomial expansions, but I don't know how to compare the terms in $\displaystyle x^{2r}$
Note that $\displaystyle (1+x)^n (1-x)^n = (1-{x^2})^n$. So multiply the two expansions on the left hand side and compare the resulting coefficient of $\displaystyle x^{2r}$ with the coefficient of $\displaystyle x^{2r}$ in the expansion on the right hand side.