Distinct Subsequences

Problem

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.

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Idea

动态规划，跪了。
动态规划，设dp[i][j]是从字符串S[0...i]中删除几个字符得到字符串T[0...j]的不同的删除方法种类，有上面递归的分析可知，动态规划方程如下

如果S[i] = T[j], dp[i][j] = dp[i-1][j-1]+dp[i-1][j]

如果S[i] 不等于 T[j], dp[i][j] = dp[i-1][j]

初始条件：当T为空字符串时，从任意的S删除几个字符得到T的方法为1

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Solution

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