Devise an algorithm to compute the number of distinct optimal alignments

$($the number of the optimal paths in alignment edit graph$)$ between a pair of DNA sequences. A replacement $($mismatch$)/$insertion$/$deletion will cost $1.$ A match will cost $0.$ Note that there may be multiple optimal alignments. Note that the trace$-$back is not needed. The output is a number. Please write down the recursive functions and describe the notations you use. Please also write down your pseudo$-$code. Note that the trace$-$back is not needed for this problem.