Number of Dice Rolls With Target Sum

You have n dice, and each dice 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.

Solution

var numRollsToTarget = function(n, k, target) {
    const MOD = 1000000007;

    let dp = new Array(target + 1).fill(0);
    dp[0] = 1;

    for (let i = 1; i <= n; i++) {
        const next = new Array(target + 1).fill(0);

        for (let sum = 1; sum <= target; sum++) {
            for (let face = 1; face <= k && face <= sum; face++) {
                next[sum] = (next[sum] + dp[sum - face]) % MOD;
            }
        }

        dp = next;
    }

    return dp[target];
};