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:

0 <= K < N <= L <= 100

Difficulty: MEDIUM
Tags: Dynamic programming

Combinatorics

Mathematics
Skill types: Dynamic programming

Starter code:
class Solution:
    def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
        

Solutions:
import math

class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		s = 0
		c = 0
		r = 0
		x = math.factorial(N)
		while True:
			c = x * (N - r - K) ** (L - K) * (-1) ** r // (math.factorial(N - r - K) * math.factorial(r))
			if c != 0:
				s = (s + c) % (10 ** 9 + 7)
				r += 1
			else:
				return s


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		MOD = 10 ** 9 + 7

		@lru_cache(None)
		def dp(i, j):
			if i < j:
				return 0
			if i == 0:
				return 1 if j == 0 else 0
			a = dp(i - 1, j - 1) * (N - j + 1)
			a += dp(i - 1, j) * (j - K if j > K else 0)
			return a % MOD
		return dp(L, N)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		mod = 10 ** 9 + 7

		def memo(f):
			dic = {}

			def f_alt(*args):
				if args not in dic:
					dic[args] = f(*args)
				return dic[args]
			return f_alt

		@memo
		def play(N, L):
			if L == 0:
				return 1 if N == 0 else 0
			if N > L:
				return 0
			return (N * play(N - 1, L - 1) + max(0, N - K) * play(N, L - 1)) % mod
		return play(N, L)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		dp = [[0] * (N + 1) for _ in range(L + 1)]
		dp[0][0] = 1
		for i in range(1, L + 1):
			for j in range(1, N + 1):
				dp[i][j] = dp[i - 1][j - 1] * (N - j + 1)
				if j > K:
					dp[i][j] += dp[i - 1][j] * (j - K)
		return dp[-1][-1] % (10 ** 9 + 7)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		mod = 10 ** 9 + 7
		dp = [[0] * (L + 1) for _ in range(N + 1)]
		for i in range(1, N + 1):
			for j in range(i, L + 1):
				if i == K + 1 or i == j:
					dp[i][j] = math.factorial(i)
				else:
					dp[i][j] = dp[i - 1][j - 1] * i
					if j > i:
						dp[i][j] += dp[i][j - 1] * (i - K)
				dp[i][j] %= mod
		return dp[N][L]


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		dp = [1] * (L - N + 1)
		for p in range(2, N - K + 1):
			for i in range(1, L - N + 1):
				dp[i] += dp[i - 1] * p
		ans = dp[-1]
		for k in range(2, N + 1):
			ans *= k
		return ans % (10 ** 9 + 7)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		mod = 10 ** 9 + 7

		@lru_cache(None)
		def helper(i, notplayed):
			nonlocal mod
			if i == L + 1:
				return 0 if notplayed != 0 else 1
			ans = max(N - notplayed - K, 0) * helper(i + 1, notplayed) % mod
			if notplayed != 0:
				ans += notplayed * helper(i + 1, notplayed - 1)
			return ans % mod
		return helper(1, N)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		BIG = 10 ** 9 + 7

		@lru_cache(None)
		def dp(r, n):
			if r == 0:
				return 1 if n == 0 else 0
			return (dp(r - 1, n - 1) * (N - (n - 1)) + dp(r - 1, n) * max(0, n - K)) % BIG
		return dp(L, N)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:

		@lru_cache(None)
		def unique(length, uni):
			if uni == 0:
				return 0
			if length == 1:
				if uni == 1:
					return N
				else:
					return 0
			ret = unique(length - 1, uni - 1) * (N - uni + 1)
			ret += unique(length - 1, uni) * max(0, uni - K)
			return ret % (10 ** 9 + 7)
		return unique(L, N)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		mod = 10 ** 9 + 7

		@lru_cache(None)
		def dp(l, n):
			if not l:
				return not n
			return dp(l - 1, n - 1) * (N - n + 1) + dp(l - 1, n) * max(n - K, 0)
		return dp(L, N) % mod


from functools import lru_cache

class Solution:

	def numMusicPlaylists(self, N, L, K):

		@lru_cache(None)
		def dp(i, j):
			if i == 0:
				return +(j == 0)
			ans = dp(i - 1, j - 1) * (N - j + 1)
			ans += dp(i - 1, j) * max(j - K, 0)
			return ans % (10 ** 9 + 7)
		return dp(L, N)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		dp = [[0 for i in range(L + 1)] for j in range(N + 1)]
		for i in range(1, len(dp)):
			for j in range(1, len(dp[0])):
				if i == j:
					dp[i][j] = math.factorial(i)
				else:
					dp[i][j] = dp[i - 1][j - 1] * i + dp[i][j - 1] * max(i - K, 0)
		print(dp)
		return dp[N][L] % (10 ** 9 + 7)


import math

class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		mod = 10 ** 9 + 7
		dp = [[0 for _ in range(L + 1)] for _ in range(N + 1)]
		dp[0][0] = 1
		for i in range(1, N + 1):
			for j in range(1, L + 1):
				dp[i][j] = dp[i - 1][j - 1] * (N - i + 1) % mod
				dp[i][j] += dp[i][j - 1] * max(i - K, 0) % mod
		return dp[-1][-1] % mod


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		memo = {}

		def dp(i, j):
			if i == 0:
				return j == 0
			if (i, j) in memo:
				return memo[i, j]
			memo[i, j] = dp(i - 1, j - 1) * (N - j + 1) + dp(i - 1, j) * max(j - K, 0)
			return memo[i, j]
		return dp(L, N) % (10 ** 9 + 7)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:

		@lru_cache(None)
		def dp(i, j):
			if i == 0:
				return j == 0
			return (dp(i - 1, j) * max(0, j - K) + dp(i - 1, j - 1) * (N - j + 1)) % (10 ** 9 + 7)
		return dp(L, N)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		dp = [[0 for n in range(N + 1)] for l in range(L + 1)]
		dp[0][0] = 1
		for l in range(1, L + 1):
			for n in range(1, N + 1):
				dp[l][n] += dp[l - 1][n - 1] * (N - n + 1)
				dp[l][n] += dp[l - 1][n] * max(n - K, 0)
				dp[l][n] = dp[l][n] % (10 ** 9 + 7)
		return dp[L][N]


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		dp = [[0] * (N + 1) for _ in range(L + 1)]
		dp[0][0] = 1
		for l in range(1, L + 1):
			for n in range(1, N + 1):
				dp[l][n] += dp[l - 1][n - 1] * (N - n + 1)
				dp[l][n] += dp[l - 1][n] * max(n - K, 0)
				dp[l][n] = dp[l][n] % 1000000007
		return dp[L][N]


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		mod = 10 ** 9 + 7
		memo = {}

		def helper(i, j):
			nonlocal N, K
			if i == 0:
				if j == 0:
					return 1
				else:
					return 0
			if (i, j) in memo:
				return memo[i, j]
			ans = 0
			ans += helper(i - 1, j - 1) * (N - (j - 1))
			ans += helper(i - 1, j) * max(0, j - K)
			memo[i, j] = ans % mod
			return ans % mod
		return helper(L, N)


class Solution:

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:

		@lru_cache(None)
		def dp(unique, total, r):
			if total == L:
				return 1
			if unique <= K:
				return r * dp(unique + 1, total + 1, r - 1)
			else:
				ans = 0
				if r < L - total:
					ans += (unique - K) * dp(unique, total + 1, r)
				ans += r * dp(unique + 1, total + 1, r - 1)
				return ans
		return dp(0, 0, N) % (10 ** 9 + 7)


class Solution:
	import math
	from functools import lru_cache

	def numMusicPlaylists(self, N: int, L: int, K: int) -> int:
		MOD = 1000000007

		@lru_cache(maxsize=None)
		def rec(n, l):
			if l < n or n <= K:
				return 0
			elif l == n:
				return math.factorial(n) % MOD
			return (n * rec(n - 1, l - 1) + (n - K) * rec(n, l - 1)) % MOD
		return rec(N, L)