920. Number of Music Playlists

Your music player contains N different songs and she wants to listen to L (not necessarily different) songs during your trip.  You create a playlist so that:

• Every song is played at least once
• A song can only be played again only if K other songs have been played

Return the number of possible playlists.  As the answer can be very large, return it modulo 10^9 + 7.

Example 1:

Input: N = 3, L = 3, K = 1
Output: 6
Explanation: There are 6 possible playlists. [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1].

Example 2:

Input: N = 2, L = 3, K = 0
Output: 6
Explanation: There are 6 possible playlists. [1, 1, 2], [1, 2, 1], [2, 1, 1], [2, 2, 1], [2, 1, 2], [1, 2, 2]

Example 3:

Input: N = 2, L = 3, K = 1
Output: 2
Explanation: There are 2 possible playlists. [1, 2, 1], [2, 1, 2]

Note:

1. 0 <= K < N <= L <= 100

Approach #1: DP. [C++]

class Solution {
    public int numMusicPlaylists(int N, int L, int K) {
        int mod = (int)Math.pow(10, 9) + 7;
        long[][] dp = new long[L+1][N+1];
        dp[0][0] = 1;
        for (int i = 1; i <= L; ++i) {
            for (int j = 1; j <= N; ++j) {
                dp[i][j] = (dp[i-1][j-1] * (N - (j - 1))) % mod;
                if (j > K) {
                    dp[i][j] = (dp[i][j] + (dp[i-1][j] * (j - K)) % mod) % mod;
                }
            }
        }
        return (int)dp[L][N];
    }
}

　　

Analysis:

dp[i][j] denotes the solution of i songs with j difference songs. So the final answer should be dp[L][N]

Think one step before the last one, there are only cases for the answer of dp[i][j]

case 1 (the last added one is new song): listen i - 1 songs with j - 1 difference songs, then the last one is definitely new song with the choices of N - (j - 1).

case2 (the last added one is old song): listen i - 1 songs with j different songs, then the last one is definitely old song with the choices of j if without the constraint of K, the status equation will be dp[i][j] = dp[i-1][j-1] * (N - (j - 1)) + dp[i-1][j] * j

If with the constaint of K, there are also two cases 

Case 1: no changes since the last added one is new song. Hence, there is no conflict 

Case 2: now we don't have choices of j for the last added old song. Itt should be updateed j - k because k songs can't be chsed from j - 1 to j - k. However, if j <= K, this case will be 0 because only after choosing K different other songs, old song can be chosen.

if (j > k)

dp[i][j] = dp[i-1][j-1] * (N-(j-1)) + dp[i-1][j] * (j-k)

else

dp[i][j] = dp[i-1][j-1] * (N - (j-1))

Reference:

https://leetcode.com/problems/number-of-music-playlists/discuss/180338/DP-solution-that-is-Easy-to-understand