# 2902. Count of Sub-Multisets With Bounded Sum

## Description

You are given a 0-indexed array nums of non-negative integers, and two integers l and r.

Return the count of sub-multisets within nums where the sum of elements in each subset falls within the inclusive range of [l, r].

Since the answer may be large, return it modulo 109 + 7.

A sub-multiset is an unordered collection of elements of the array in which a given value x can occur 0, 1, ..., occ[x] times, where occ[x] is the number of occurrences of x in the array.

Note that:

• Two sub-multisets are the same if sorting both sub-multisets results in identical multisets.
• The sum of an empty multiset is 0.

Example 1:

Input: nums = [1,2,2,3], l = 6, r = 6
Output: 1
Explanation: The only subset of nums that has a sum of 6 is {1, 2, 3}.


Example 2:

Input: nums = [2,1,4,2,7], l = 1, r = 5
Output: 7
Explanation: The subsets of nums that have a sum within the range [1, 5] are {1}, {2}, {4}, {2, 2}, {1, 2}, {1, 4}, and {1, 2, 2}.


Example 3:

Input: nums = [1,2,1,3,5,2], l = 3, r = 5
Output: 9
Explanation: The subsets of nums that have a sum within the range [3, 5] are {3}, {5}, {1, 2}, {1, 3}, {2, 2}, {2, 3}, {1, 1, 2}, {1, 1, 3}, and {1, 2, 2}.

Constraints:

• 1 <= nums.length <= 2 * 104
• 0 <= nums[i] <= 2 * 104
• Sum of nums does not exceed 2 * 104.
• 0 <= l <= r <= 2 * 104

## Solutions

• class Solution:
def countSubMultisets(self, nums: List[int], l: int, r: int) -> int:
kMod = 1_000_000_007
# dp[i] := # of submultisets of nums with sum i
dp = [1] + [0] * r
count = collections.Counter(nums)
zeros = count.pop(0, 0)

for num, freq in count.items():
# stride[i] := dp[i] + dp[i - num] + dp[i - 2 * num] + ...
stride = dp.copy()
for i in range(num, r + 1):
stride[i] += stride[i - num]
for i in range(r, 0, -1):
if i >= num * (freq + 1):
# dp[i] + dp[i - num] + dp[i - freq * num]
dp[i] = stride[i] - stride[i - num * (freq + 1)]
else:
dp[i] = stride[i]

return (zeros + 1) * sum(dp[l : r + 1]) % kMod


• 
class Solution {
static final int MOD = 1_000_000_007;
public int countSubMultisets(List<Integer> nums, int l, int r) {
Map<Integer, Integer> count = new HashMap<>();
int total = 0;
for (int num : nums) {
total += num;
if (num <= r) {
count.merge(num, 1, Integer::sum);
}
}
if (total < l) {
return 0;
}
r = Math.min(r, total);
int[] dp = new int[r + 1];
dp[0] = count.getOrDefault(0, 0) + 1;
count.remove(Integer.valueOf(0));
int sum = 0;
for (Map.Entry<Integer, Integer> e : count.entrySet()) {
int num = e.getKey();
int c = e.getValue();
sum = Math.min(sum + c * num, r);
// prefix part
// dp[i] = dp[i] + dp[i - num] + ... + dp[i - c*num] + dp[i-(c+1)*num] + ... + dp[i %
// num]
for (int i = num; i <= sum; i++) {
dp[i] = (dp[i] + dp[i - num]) % MOD;
}
int temp = (c + 1) * num;
// correction part
// subtract dp[i - (freq + 1) * num] to the end part.
// leves dp[i] = dp[i] + dp[i-num] +...+ dp[i - c*num];
for (int i = sum; i >= temp; i--) {
dp[i] = (dp[i] - dp[i - temp] + MOD) % MOD;
}
}
int ans = 0;
for (int i = l; i <= r; i++) {
ans += dp[i];
ans %= MOD;
}
return ans;
}
}