1155. Number of Dice Rolls With Target Sum

Medium

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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

class Solution:
    def numRollsToTarget(self, n: int, k: int, target: int) -> int:
        mem = {}
        MOD = 10**9+7
        def dp(n, target):
            nonlocal k
            if n == 0:
                return 0 if target > 0 else 1
            key = (n, target)
            if key in mem:
                return mem[key]
            total = 0
            for i in range(max(0, target-k), target):
                total += dp(n-1, i)
            mem[key] = total
            return total
        return dp(n, target) % MOD