This is a homework problem, please don't blurt out the answer! 🙂
I've been given the following, and asked to evaluate the sum:
$$\sum_{k = 0}^{n}(-1)^k\binom{n}{k}10^k$$
So, I started out trying to look at this as equivalent to the binomial theorem, in which case, I could attempt something like this: $10^k = y^{n-k}$ but I didn't feel that got me anywhere.
So I started actually evaluating it…
$$(-1)^0\binom{n}{0}10^0 + (-1)^1\binom{n}{1}10^1 + \ldots + (-1)^n\binom{n}{n}10^n$$
So, if I'm thinking correctly, all the other terms cancel out and you are left with:
$$(-1)^n\binom{n}{n}10^n = (-1)^n10^n$$
But, obviously this cannot be correct (or can it?). The book gives a slightly different answer, so I'm wondering where I'm going wrong. Some direction would be greatly appreciated!
Books answer: $\displaystyle (-1)^n9^n$
Best Answer
Hint: If you're evaluating $(a+b)^n$, what does the binomial theorem say this is equal to and how do you relate that to the sum you're given?