2003 AMC 12B 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 \qquad\mathrm{(B)}\ 6 \qquad\mathrm{(C)}\ 11 \qquad\mathrm{(D)}\ 16 \qquad\mathrm{(E)}\ 19

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.