Number of Dice Rolls With Target Sum

You have n dice and each die has k faces numbered from 1 to k.

Given three integers n, k, and target, return the number of possible ways (out of the kn total ways) to roll the dice so the sum of the face-up numbers equals target. Since the answer may be too large, return it modulo 109 + 7.

Example 1:

Input: n = 1, k = 6, target = 3
Output: 1
Explanation: You throw one die with 6 faces.
There is only one way to get a sum of 3.

Example 2:

Input: n = 2, k = 6, target = 7
Output: 6
Explanation: You throw two dice, each with 6 faces.
There are 6 ways to get a sum of 7: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1.

Example 3:

Input: n = 30, k = 30, target = 500
Output: 222616187
Explanation: The answer must be returned modulo 109 + 7.

Constraints:

• 1 <= n, k <= 30
• 1 <= target <= 1000

SOLUTION:

from functools import lru_cache

class Solution:
    @lru_cache(maxsize=None)
    def numRollsToTarget(self, n: int, k: int, target: int) -> int:
        if n == 1:
            return 1 if 1 <= target <= k else 0
        ways = 0
        for i in range(1, k + 1):
            ways += self.numRollsToTarget(n - 1, k, target - i)
        return ways % (10 ** 9 + 7)