how to find the number of distinct subsequences of a string?

It's a classic dynamic programming problem.

Let:

dp[i] = number of distinct subsequences ending with a[i]
sum[i] = dp[1] + dp[2] + ... + dp[i]. So sum[n] will be your answer.
last[i] = last position of character i in the given string.

A null string has one subsequence, so dp[0] = 1.

read a
n = strlen(a)
for i = 1 to n
  dp[i] = sum[i - 1] - sum[last[a[i]] - 1]
  sum[i] = sum[i - 1] + dp[i]
  last[a[i]] = i

return sum[n]

Explanation

dp[i] = sum[i - 1] - sum[last[a[i]] - 1]

Initially, we assume we can append a[i] to all subsequences ending on previous characters, but this might violate the condition that the counted subsequences need to be distinct. Remember that last[a[i]] gives us the last position a[i] appeared on until now. The only subsequences we overcount are those that the previous a[i] was appended to, so we subtract those.

sum[i] = sum[i - 1] + dp[i]
last[a[i]] = i

Update these values as per their definition.

If your indexing starts from 0, use a[i - 1] wherever I used a[i]. Also remember to wrap your computations in a mod function if you're going to submit code. This should be implemented like this:

mod(x) = (x % m + m) % m

In order to correctly handle negative values in some languages (such as C/C++).

There exists an easier solution to this problem.

The idea is: If all character of the string are distinct, total number of subsequences is 2^n. Now, if we find any character that have already occurred before, we should consider its last occurrence only (otherwise sequence won't be distinct). So we have to subtract the number of subsequences due to its previous occurrence.

My implementation is like this:

read s
dp[0] = 1
len = strlen(s)
last[s.length()] = {-1} //declaring `last` array with same as length of string `s` and all elements initialized with -1.

for (i = 1; i <= len; i++) 
{
    dp[i] = (dp[i - 1] * 2)
    if (last[s[i]] > 0) dp[i] = (dp[i] - dp[last[s[i]] - 1])
    last[s[i]] = i
}