<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:
    def numMusicPlaylists(self, n: int, goal: int, k: int) -> int:
        # dp[j]: # of permutation if we've selected j unique songs out of x songs selected,
        # where x depends on which iteration we are currently at
        dp = [0] * (n + 1)

        # basecase: when x == goal -> then # of perm = 1 only if j = n
        dp[-1] = 1

        # tabulate
        for n_pick in range(goal - 1, -1, -1):
            for n_unique in range(n + 1):
                # no way n_unique > n_pick
                if n_unique > n_pick:
                    break

                # select old song
                if n_unique > k:
                    dp[n_unique] *= (n_unique - k) 
                else:
                    dp[n_unique] = 0

                # select new song
                if n > n_unique:
                    dp[n_unique] += dp[n_unique + 1] * (n - n_unique)
            
                dp[n_unique] %= (10 ** 9 + 7)
        
        return dp[0]

                
                

                
16500