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We define a dynamic programming approach where we use a 2D DP table where dp[i][j] means the number of ways to form a playlist of length j using i distinct songs. We will eventually find dp[n][goal].
The key recurrence relations are:
(n - i + 1) songs available, hence dp[i][j] += dp[i - 1][j - 1] * (n - i + 1)i songs, considering k songs must be between repeats, we have: dp[i][j] += dp[i][j - 1] * (i - k), where i > kThis draws from the constraints on repeats and expands the playlist iteratively.
Time Complexity: O(n * goal), since we iterate through each subproblem.
Space Complexity: O(n * goal), due to the size of the DP table.
1
2MOD = 10**9 + 7
3def numMusicPlaylists(n, goal, k):
4 dp = [[0] * (goal + 1) for _ in range(n + 1)]
5 dp[0][0] = 1
6 for i in range(1, n + 1):
7 for j in range(1, goal + 1):
8 dp[i][j] = dp[i - 1][j - 1] * (n - i + 1) % MOD
9 if i > k:
10 dp[i][j] += dp[i][j - 1] * (i - k) % MOD
11 dp[i][j] %= MOD
12 return dp[n][goal]
13The solution uses a 2D list to store the number of ways to create a playlist of length j with i distinct songs. The state transition involves adding a new song or reusing one respecting the separation condition. The final solution is found at dp[n][goal].
This approach uses memoization to recursively compute the number of playlists. It calculates the playlists by considering whether a new song is added or an existing song is reused, accounting for k intervening songs.
Time Complexity: O(n * goal), due to memoization.
Space Complexity: O(n * goal), for memo storage.
1
2const numMusicPlaylists = (n, goal
The JavaScript solution implements a top-down dynamic programming approach using memoization to efficiently explore all possible playlists and caches intermediate results to optimize repeated subproblem solutions.