Dice Throws Problem Statement

You are given D dice, each having F faces numbered from 1 to F. The task is to determine the number of possible ways to roll all the dice such that the sum of the face-up numbers equals the given 'target' sum S.

Example:

Input:

2
3 6 8
2 4 5

Output:

21
4

Explanation:

In the first test case, there are 3 dice each with 6 faces and the possible ways to roll them to get a sum of 8 are 21. In the second test case, there are 2 dice each with 4 faces, and there are 4 ways to roll them to reach the sum of 5.

Constraints:

1 ≤ T ≤ 5

1 ≤ D, F ≤ 50

1 ≤ S ≤ 10^3

Note:

As the result can be large, return your answer modulo 10^9 + 7.

Follow Up:

Can you optimize the solution to use no more than O(S) extra space?

Question

Dice Throws Problem Statement

You are given D dice, each having F faces numbered from 1 to F. The task is to determine the number of possible ways to roll all the dice such that the sum of the face-up numbers equals the given 'target' sum S.

Example:

Input:

2
3 6 8
2 4 5

Output:

21
4

Explanation:

In the first test case, there are 3 dice each with 6 faces and the possible ways to roll them to get a sum of 8 are 21. In the second test case, there are 2 dice each with 4 faces, and there are 4 ways to roll them to reach the sum of 5.

Constraints:

1 ≤ T ≤ 5
1 ≤ D, F ≤ 50
1 ≤ S ≤ 10^3

Note:

As the result can be large, return your answer modulo 10^9 + 7.

Follow Up:

Can you optimize the solution to use no more than O(S) extra space?

Accepted Answer

The task is to determine the number of possible ways to roll all the dice such that the sum of the face-up numbers equals the given 'target' sum.

Use dynamic programming to solve the problem efficiently.
Create a 2D array to store the number of ways to achieve each sum with different number of dice.
Iterate through the dice and sum possibilities to fill up the array.
Return the result modulo 10^9 + 7.
Optimize the solution to use no more than O(S) extra space by using rolling arrays.