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Solution: Count the Number of Good Subsequences

Explore how to efficiently count good subsequences in a string by applying dynamic programming concepts. This lesson guides you through calculating character frequencies, using binomial coefficients for combinations, and employing memoization to optimize factorial and modular inverse computations. By the end, you will understand how to handle the problem with time and space complexity of O(n), enabling you to solve similar subsequence problems effectively.

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Statement

Count and return the number of good subsequences in the given string s. You may return the modulo 109+710^9 + 7 of the count.

  • A subsequence is a sequence formed from another sequence by deleting some or no elements while keeping the order of the remaining elements unchanged.

  • A good subsequence is a subsequence of a string if it is not empty and the frequency of each character is the same.

Constraints:

  • 1≤1 \leqs.length ≤104\leq 10^4

  • s will only contain lowercase English characters.

Solution

The solution to this problem involves calculating the combinations. We will start by calculating the frequency of each character and then iterate over the frequencies, from 1 to the highest frequency of any character. We will count subsequences where each character appears for some or all of its frequency, i.e., calculating combinations. The count is summed up in each iteration, and after the loop terminates, 1 is deducted from the final count. The reason for deducting 1 is that the empty subsequence will also be counted. Because empty subsequence does not qualify as a good subsequence, we will remove its count from the final calculation.

Computing the combinations is a typical Binomial Coefficient problemThe binomial coefficient, often denoted as "n choose k," calculates the number of ways to choose k elements from a set of n distinct elements., known as n choose k problem, which is in mathematics written as (nk)=n!k!⋅(n−k)!\binom{n}{k} = \frac{n!}{k! \cdot (n - k)!} ...