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.

//这里可以用dfs,但是，TLE.所以，采用dp来解决.
//假设dp[i][j]的状态表示为字符串i变换到字符串j的方法的数量，那么，动态转移方程为：
// 1) S[i-1] == T[j-1] --> dp[i][j] = dp[i-1][j-1] + dp[i-1][j]
// 2) S[i-1] != T[j-1] --> dp[i][j] = dp[i-1][j].
class Solution {
public:
    int numDistinct(std::string S, std::string T) {
        int n = S.size(),m = T.size();
		std::vector<std::vector<int>> dp(n+1,std::vector<int>(m+1,0));
		for (int i = 0; i < n+1; i++)
		{
			dp[i][0] = 1;
		}
		for (int i = 1; i <= n; i++)
		{
			for (int j = 1; j <= m; j++)
			{
				if(S[i-1] == T[j-1]) dp[i][j] = dp[i-1][j-1] + dp[i-1][j];
				else dp[i][j] = dp[i-1][j];
			}
		}
		return dp[n][m];
    }
};

leetcode - Distinct Subsequences