Given a string S and a string T, count the number of distinct subsequences of T in S. A subsequence of a string is a new string which is formed from the original string by deleting some (can be none) of the characters without disturbing the relative positions of the remaining characters. (ie, “ACE” is a subsequence of “ABCDE” while “AEC” is not).

Here is an example: S = “rabbbit”, T = “rabbit” Return 3.

r a b b b i t
1 1 1 1 1 1 1 1
r 0 1 1 1 1 1 1 1
a 0 1 1 1 1 1 1 1
b 0 0 0 1 2 3 3 3
b 0 0 0 0 1 3 3 3
i 0 0 0 0 0 0 3 3
t 0 0 0 0 0 0 0 3

class Solution {
public:
int numDistinct(string S, string T) {
int dp[T.length() + 1][S.length() + 1];
for(int i = 0; i <= T.length(); i++) {
for(int j = 0; j <= S.length(); j++) {
if(i == 0) dp[i][j] = 1;
else if(j == 0) dp[i][j] = 0;
else {
if(T[i - 1] == S[j - 1]) dp[i][j] = dp[i][j - 1] + dp[i - 1][j - 1];
else dp[i][j] = dp[i][j - 1];
}
}
}

return dp[T.length()][S.length()];
}
};