The maximum number of happy groups is the maximum number of partitions you can split the groups into such that the sum of group sizes in each partition is 0 mod batchSize. At most one partition is allowed to have a different remainder (the first group will get fresh donuts anyway).

Suppose you have an array freq of length k where freq[i] = number of groups of size i mod batchSize. How can you utilize this in a dp solution?

Make a DP state dp[freq][r] that represents "the maximum number of partitions you can form given the current freq and current remainder r". You can hash the freq array to store it more easily in the dp table.

For each i from 0 to batchSize-1, the next DP state is dp[freq`][(r+i)%batchSize] where freq` is freq but with freq[i] decremented by 1. Take the largest of all of the next states and store it in ans. If r == 0, then return ans+1 (because you can form a new partition), otherwise return ans (continuing the current partition).