I came across this excercise in an old exam (in discrete math), and I don't know how to approach it: $\sum_{k=0}^{10}\left(\frac{1}{2}\right)^k\left(-1\right)^k\binom{10}{k}$ I know the answer is $2^{-10}$, but I don't know why. When I was going through my text book I saw something similar regarding Catalan numbers generating functions. Thanks!
Calculating a binomial sum
3
$\begingroup$
discrete-mathematics
binomial-coefficients
generating-functions
2 Answers
11
Hints: $(-1)^k=(-1)^{10-k}$ and binomial theorem.
-
0@Marc: $(1/2-1)^{10}$ is the form in my answer, $(-1/2+1)^{10}$ is the form in my first comment. – 2012-03-12
3
Another Hint
$\sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k} = (x+y)^n$
-
0Just realized that anon mentioned Binomial Theorem (and here is my silly hint) – 2012-03-12