Guessing

Lets start with rabbbit and rabbit.

To make the problem easier, start from one char, distinct subsequences of rabbbit and r.

distinct subsequences below, by eyeball:

• r and r has 1 distinct subsequences
• a and r has 0 distinct subsequences

These two are easy to be understood.

• ra and r has 1 distinct subsequences
• rab and r has 1 distinct subsequences
• rabbbbbbbbbbbb and r has 1 distinct subsequences

These three show that r plus any junk chars will not change distinct subsequences

• rr and r has 2 distinct subsequences
• rar and r has 2 distinct subsequences
• rara and r has 2 distinct subsequences
• rarar and r has 3 distinct subsequences

rar and r is same as rr and r, because a there has nothing to do with the number of distinct subsequences, it is a junk like above.

In another word, rar and r can be converted to ra + r and r, that is 1 + 1.

Put them into table

the value P[i][j] means that S[0..i] and T[0..j] has P[i][j] distinct subsequences.

the table with only one char

+---+---+---+---+---+---+---+---+
|   | r | a | b | b | b | i | t |
+---+---+---+---+---+---+---+---+
| r | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+---+---+---+---+---+---+---+---+

Enlarge the table

+---+---+---+---+---+---+---+---+
|   | r | a | b | b | b | i | t |
+---+---+---+---+---+---+---+---+
| r | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+---+---+---+---+---+---+---+---+
| a |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
| b |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
| b |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
| i |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
| t |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+

Some grids here are easy to be filled, the grids which j > i. That is, T is longer than S, there will be no distinct subsequences.

+---+---+---+---+---+---+---+---+
|   | r | a | b | b | b | i | t |
+---+---+---+---+---+---+---+---+
| r | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+---+---+---+---+---+---+---+---+
| a | 0 |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
| b | 0 | 0 |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
| b | 0 | 0 | 0 |   |   |   |   |
+---+---+---+---+---+---+---+---+
| i | 0 | 0 | 0 | 0 |   |   |   |
+---+---+---+---+---+---+---+---+
| t | 0 | 0 | 0 | 0 | 0 |   |   |
+---+---+---+---+---+---+---+---+

j == i grids are easy too, the S and T must be the same if it wants to have 1 distinct subsequences.

+---+---+---+---+---+---+---+---+
|   | r | a | b | b | b | i | t |
+---+---+---+---+---+---+---+---+
| r | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+---+---+---+---+---+---+---+---+
| a | 0 | 1 |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
| b | 0 | 0 | 1 |   |   |   |   |
+---+---+---+---+---+---+---+---+
| b | 0 | 0 | 0 | 1 |   |   |   |
+---+---+---+---+---+---+---+---+
| i | 0 | 0 | 0 | 0 | 0 |   |   |
+---+---+---+---+---+---+---+---+
| t | 0 | 0 | 0 | 0 | 0 | 0 |   |
+---+---+---+---+---+---+---+---+

This pictrue is just encouraging you, as there is only a few grids left to be filled.

rab and ra

This problem looks familiar. It is the ra and r.

as b is a junk char, rab should have the same number as ra and ra.

Then the table is like

+---+---+---+---+---+---+---+---+
|   | r | a | b | b | b | i | t |
+---+---+---+---+---+---+---+---+
| r | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+---+---+---+---+---+---+---+---+
| a | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
+---+---+---+---+---+---+---+---+
| b | 0 | 0 | 1 |   |   |   |   |
+---+---+---+---+---+---+---+---+
| b | 0 | 0 | 0 | 1 |   |   |   |
+---+---+---+---+---+---+---+---+
| i | 0 | 0 | 0 | 0 | 0 |   |   |
+---+---+---+---+---+---+---+---+
| t | 0 | 0 | 0 | 0 | 0 | 0 |   |
+---+---+---+---+---+---+---+---+

rabb and rab

This time, add a new b to both rab and ra.

Use the new b in subsequences

This time they have same letter, b , at the end. It is not a junk.

This time, remove b from both S and T, then they become rab and ra.

that is, this will take rab and ra distinct subsequences.

Dont use the new b

Treat b as junk, it has rab and rab distinct subsequences.

Sum up two choices

Use or not will result two set of distinct subsequences, so sum them up.

rabb and rab = rab and ra + rab and rab.

+---+---+---+---+---+---+---+---+
|   | r | a | b | b | b | i | t |
+---+---+---+---+---+---+---+---+
| r | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+---+---+---+---+---+---+---+---+
| a | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
+---+---+---+---+---+---+---+---+
| b | 0 | 0 | 1 | 2 |   |   |   |
+---+---+---+---+---+---+---+---+
| b | 0 | 0 | 0 | 1 |   |   |   |
+---+---+---+---+---+---+---+---+
| i | 0 | 0 | 0 | 0 | 0 |   |   |
+---+---+---+---+---+---+---+---+
| t | 0 | 0 | 0 | 0 | 0 | 0 |   |
+---+---+---+---+---+---+---+---+

General form

the grid P[i][j] has 1 choice when S[i] != T[j], this only adds some junk chars.

P[i][j] = P[i - 1][j] 

the grid P[i][j] has 2 choice when S[i] == T[j]

P[i][j] = 
P[i - 1][j] // adds as junk chars.
+
P[i - 1][j - 1] // use the new char in the new subsequence

Fill up the table and the bottom right grid is the answer.