Evaluate the sum ${n \choose 1} + 3{n \choose 3} +5{n \choose 5} + 7{n \choose 7}...$ in closed form How do I evaluate the sum:$${n \choose 1} + 3{n \choose 3} +5{n \choose 5} + 7{n \choose 7}  ...$$in closed form?
I don't really know how to start and approach this question. Any help is greatly appreciated.

Here is a combinatorial proof because I like combinatorial proofs.
Note that that $\sum_{k \geq 0} \binom{n}{2k} = 2^{n-1}$. One way to prove this result is with the binomial theorem. We will use this fact.
Now $\sum_{k \geq 0} (2k+1)\binom{n}{2k+1}$ can be interpreted as choosing a subset of odd size from a set with $n$ elements in one of $\binom{n}{2k+1}$ ways and then choosing a distinguished element from such a subset in one of $2k+1$ ways. 
Another way to make the same choice would be to first choose the distinguished element in one of $n$ ways and then choose the remaining elements of such a subset in one of $\binom{n-1}{2k}$ ways. Thus we have
$$\sum_{k \geq 0} (2k+1)\binom{n}{2k+1} =  n \sum_{k\geq 0} \binom{n-1}{2k} = n 2^{n-2}.$$
Notice this solution essentially uses the hint Did gave in the comments. The identity Did provides can be proven combinatorially as we have or algebraically.
