<p>You have <code>n</code> dice, and each dice has <code>k</code> faces numbered from <code>1</code> to <code>k</code>.</p>

<p>Given three integers <code>n</code>, <code>k</code>, and <code>target</code>, return <em>the number of possible ways (out of the </em><code>k<sup>n</sup></code><em> total ways) </em><em>to roll the dice, so the sum of the face-up numbers equals </em><code>target</code>. Since the answer may be too large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

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

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 2, k = 6, target = 7
<strong>Output:</strong> 6
<strong>Explanation:</strong> 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.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 30, k = 30, target = 500
<strong>Output:</strong> 222616187
<strong>Explanation:</strong> The answer must be returned modulo 10<sup>9</sup> + 7.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n, k &lt;= 30</code></li>
	<li><code>1 &lt;= target &lt;= 1000</code></li>
</ul>

class Solution {
public:
    long long solve(int dice, int face, int target, vector<vector<long long>>& dp) {
        if (target < 0) {
            return 0;
        }
        if (dice == 0 && target == 0) {
            return 1; 
        }
        if (dice == 0 || target == 0) {
            return 0; 
        }
        if (dp[dice][target] != -1) {
            return dp[dice][target]; 
        }

        long long ans = 0;
        for (int i = 1; i <= face; i++) { 
            ans = (ans + solve(dice - 1, face, target - i, dp)) % 1000000007;
        }
        return dp[dice][target] = ans; 
    }

    int numRollsToTarget(int n, int k, int target) {
        vector<vector<long long>> dp(n + 1, vector<long long>(target + 1, -1)); 
        return solve(n, k, target, dp);
    }
};

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