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1155. Number of Dice Rolls With Target Sum

Description

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.

 

Constraints:

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

Solutions

Solution 1: Dynamic Programming

We define $f[i][j]$ as the number of ways to get a sum of $j$ using $i$ dice. Then, we can obtain the following state transition equation:

$$ f[i][j] = \sum_{h=1}^{\min(j, k)} f[i-1][j-h] $$

where $h$ represents the number of points on the $i$-th die.

Initially, we have $f[0][0] = 1$, and the final answer is $f[n][target]$.

The time complexity is $O(n \times k \times target)$, and the space complexity is $O(n \times target)$.

We notice that the state $f[i][j]$ only depends on $f[i-1][]$, so we can use a rolling array to optimize the space complexity to $O(target)$.

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class Solution:
    def numRollsToTarget(self, n: int, k: int, target: int) -> int:
        f = [[0] * (target + 1) for _ in range(n + 1)]
        f[0][0] = 1
        mod = 10**9 + 7
        for i in range(1, n + 1):
            for j in range(1, min(i * k, target) + 1):
                for h in range(1, min(j, k) + 1):
                    f[i][j] = (f[i][j] + f[i - 1][j - h]) % mod
        return f[n][target]

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class Solution {
    public int numRollsToTarget(int n, int k, int target) {
        final int mod = (int) 1e9 + 7;
        int[][] f = new int[n + 1][target + 1];
        f[0][0] = 1;
        for (int i = 1; i <= n; ++i) {
            for (int j = 1; j <= Math.min(target, i * k); ++j) {
                for (int h = 1; h <= Math.min(j, k); ++h) {
                    f[i][j] = (f[i][j] + f[i - 1][j - h]) % mod;
                }
            }
        }
        return f[n][target];
    }
}

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class Solution {
public:
    int numRollsToTarget(int n, int k, int target) {
        const int mod = 1e9 + 7;
        int f[n + 1][target + 1];
        memset(f, 0, sizeof f);
        f[0][0] = 1;
        for (int i = 1; i <= n; ++i) {
            for (int j = 1; j <= min(target, i * k); ++j) {
                for (int h = 1; h <= min(j, k); ++h) {
                    f[i][j] = (f[i][j] + f[i - 1][j - h]) % mod;
                }
            }
        }
        return f[n][target];
    }
};

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func numRollsToTarget(n int, k int, target int) int {
    const mod int = 1e9 + 7
    f := make([][]int, n+1)
    for i := range f {
        f[i] = make([]int, target+1)
    }
    f[0][0] = 1
    for i := 1; i <= n; i++ {
        for j := 1; j <= min(target, i*k); j++ {
            for h := 1; h <= min(j, k); h++ {
                f[i][j] = (f[i][j]