I'm not sure why i'm wrong. I think is in my understanding of subsequence. Even by reading further I think my algorithm is ok, but I only get right the testcase.

My result is just this one:

result = (a * b) + (c * d) + (a * b * c * d);

Note: a, b, c and d represent the number of times that each letter appears in the original string.

first part (a * b) represents the number of different perfect subsequences with only a and b.

second part (c * d), perfect subsequences with c and d.

and third part ( a * b * c * d), the perfect subsequences with a, b, c and d.

E X A M P L E S

Example 1: abcd

a = 1, b = 1, c = 1, d = 1

perfect subsequences = ab, cd, abcd

Those are 3 perfect subsequences.

if we do (a*b)+(c*d)+(a*b*c*d), this is 1 + 1 + 1 = 3, so in here we're right!

Example 2: aabbccdd

a = 2, b = 2, c = 2, d = 2

• perfect subsequences ab:

which is the same as a * b.

• perfect subsequences cd:

which is the same as c * d.

• perfect subsequences abcd:

which is the same as (ab subsequences) * (cd subsequences)

and ab = a * b, cd = c * d, so,

is the same as a * b * c * d.

tl;dr

ab perfect subsequences = a * b,

cd perfect subsequences = c * d,

abcd perfect subsequences = a * b * c * d.

so all possible perfect subsequences = ab + cd + abcd = a * b + c * d + a * b * c * d.

But obviously I'm doing something wrong, but I don't know what. My guess is that I'm underestimating the difficulty of this problem, that it isn't so straigth forward.