HMMT 2012 C6

For a permutation $\sigma$ of $1,2, \ldots, 7$, a transposition is a swapping of two elements. (For instance, we could apply a transposition to the permutation $3,7,1,4,5,6,2$ and get $3,7,6,4,5,1,2$ by swapping the 1 and the 6.) Let $f(\sigma)$ be the minimum number of transpositions necessary to turn $\sigma$ into the permutation $1,2,3,4,5,6,7$. Find the sum of $f(\sigma)$ over all permutations $\sigma$ of $1,2, \ldots, 7$.