/sci/ - Science & Math

I'm trying to solve the following problem in Bona's 'A Walk Through Combinatorics'. I have an answer that I believe is correct but I would like someone to check it.

For a fixed k, there are [math] \binom{n}{k} [/math] k-element subsets of [n]. This will be the number of ways to pick C. For each of these ways, there are k elements we can pick from to be the element in common in between the two sets A and B. And then for each of these k elements to pick from, there are [math](2^{k-1})^2[/math] ways to pick A and B. So, putting everything together, the answer should be [math]\sum_{i=1}^n \binom{n}{k} k (2^{k-1})^2[/math].

is this correct?