<p>Your music player contains <code>n</code> different songs. You want to listen to <code>goal</code> songs (not necessarily different) during your trip. To avoid boredom, you will create a playlist so that:</p>

<ul>
	<li>Every song is played <strong>at least once</strong>.</li>
	<li>A song can only be played again only if <code>k</code> other songs have been played.</li>
</ul>

<p>Given <code>n</code>, <code>goal</code>, and <code>k</code>, return <em>the number of possible playlists that you can create</em>. Since the answer can be very 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 = 3, goal = 3, k = 1
<strong>Output:</strong> 6
<strong>Explanation:</strong> There are 6 possible playlists: [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], and [3, 2, 1].
</pre>

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

<pre>
<strong>Input:</strong> n = 2, goal = 3, k = 0
<strong>Output:</strong> 6
<strong>Explanation:</strong> There are 6 possible playlists: [1, 1, 2], [1, 2, 1], [2, 1, 1], [2, 2, 1], [2, 1, 2], and [1, 2, 2].
</pre>

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

<pre>
<strong>Input:</strong> n = 2, goal = 3, k = 1
<strong>Output:</strong> 2
<strong>Explanation:</strong> There are 2 possible playlists: [1, 2, 1] and [2, 1, 2].
</pre>

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

<ul>
	<li><code>0 &lt;= k &lt; n &lt;= goal &lt;= 100</code></li>
</ul>

class Solution {
public:
    #define ll long long
    const int MOD = 1e9 + 7;

    // Recursive brute-force dp solution
    ll bruteForceDP(int n, int goal, int k, vector<vector<int>>& dp) {
        // Base case: if no songs and goal is achieved
        if (n == 0 && goal == 0) {
            return 1;
        }
        // If either no songs are left or the goal is not met
        if (n == 0 || goal == 0) {
            return 0;
        }
        // If already computed, return the cached result
        if (dp[n][goal] != -1) {
            return dp[n][goal];
        }

        // Try to pick a new song (distinct) and add to the playlist
        ll pickNew = bruteForceDP(n - 1, goal - 1, k, dp) * n;

        // Try to replay an old song that hasn't been played in the last k songs
        ll replayOld = bruteForceDP(n, goal - 1, k, dp) * max(n - k, 0);

        // Store the result in dp and return it, modulo MOD
        return dp[n][goal] = (pickNew + replayOld) % MOD;
    }

    int numMusicPlaylists(int n, int goal, int k) {
        // Initialize dp table with -1 to indicate uncomputed states
        vector<vector<int>> dp(n + 1, vector<int>(goal + 1, -1));
        // Start the recursive brute-force dp
        return bruteForceDP(n, goal, k, dp);
    }
};

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