# [LeetCode] Distinct Subsequences (Java)

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.

## Analysis

This is a DP problem. We can use num[i][j] to save the number of distinct subsequences of T(0, j) in S(0, i). We know that for any number i, num[i][0] = 1 because we need to delete all characters from S(0, i) to a empty string.

If the character at position i in S is equal to the character at position j in T, there are two options.

1. Delete the character at position i in S. Then the number of distinct subsequences should be the number of distinct subsequences of T(0, j) in S(0, i – 1).
2. Remains the character at position i in S. Then the number is the number of distinct subsequences of T(0, j – 1) in S(0, i – 1).

So num[i][j] = num[i – 1][j] + num[i – 1][j – 1].

If the character at position i in S is not equal to the character at position j in T, then we can only delete this character. So num[i][j] = num[i – 1][j].

## Complexity

The complexity is $O(mn)$.