Question

Why does this expand to this? [xk]∑j≥0(n−1j)xj+[xk]∑j≥0(n−1j)xj+1=(n−1k)+(n−1k−1)

Answer 1

\[[x^k]\sum_{j \ge 0}^{} \left(\begin{matrix}n-1 \\ j\end{matrix}\right) x^j+[x^k]\sum_{j \ge 0}^{} \left(\begin{matrix}n-1 \\ j\end{matrix}\right) x^{j+1}= \left(\begin{matrix}n-1 \\ k\end{matrix}\right)+\left(\begin{matrix}n-1 \\ k-1\end{matrix}\right)\]

Answer 2

I wish I could help you, but I have no idea what's going on with summations, when do I learn them?

Answer 3

this is from intro to combinatorics

Answer 4

what is x here?

Answer 5

x is a variable, any real number

Answer 6

so that identity is not true...i let x=0 so \[0=\text{something}\]???

Answer 7

actually, let me check

Answer 8

x has to be nonnegative

Answer 9

also this may help \[(1+x)^n=\sum_{k=0}^{n}\left(\begin{matrix}n \\ k\end{matrix}\right)x^k\]