AMC 12 2003b Problem 19

Let $S$ be the set of permutations of the sequence $1,2,3,4,5$ for which the first term is not $1$. A permutation is chosen randomly from $S$. The probability that the second term is $2$, in lowest terms, is $a/b$. What is $a+b$?

$\mathrm{(A)}\ 5 \mathrm{(B)}\ 6 \mathrm{(C)}\ 11 \mathrm{(D)}\ 16 \mathrm{(E)}\ 19$