Question

How is one supposed to think of a solution like this?!

Answer 1

very carefully

Answer 2

That's true. Here's the problem.\[\binom{n}{k}-\binom{n}{k-1}+\binom{n}{k-2} - \binom{n}{k-3} + \cdots + (-1)^{r} \binom{n}{k-r}\]Evaluate this.

Answer 3

this reminds me of hockey stick pattern

Answer 4

you try something like this

\[\sum\limits_{i=0}^r (-1)^i*\dbinom{n}{ k-i}=\sum\limits_{i=0}^k (-1)^i \dbinom{n}{i} - \sum\limits_{i=0}^{k-r-1} (-1)^i \dbinom{n}{i}\]