LeetCode #920 — HARD

Number of Music Playlists

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

The Problem

Problem Statement

Your music player contains n different songs. You want to listen to goal songs (not necessarily different) during your trip. To avoid boredom, you will 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.

Given n, goal, and k, return the number of possible playlists that you can create. Since the answer can be very large, return it modulo 10⁹ + 7.

Example 1:

Input: n = 3, goal = 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], and [3, 2, 1].

Example 2:

Input: n = 2, goal = 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], and [1, 2, 2].

Example 3:

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

Constraints:

0 <= k < n <= goal <= 100

Step 01

Brute Force Baseline

Problem summary: Your music player contains n different songs. You want to listen to goal songs (not necessarily different) during your trip. To avoid boredom, you will 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. Given n, goal, and k, return the number of possible playlists that you can create. Since the answer can be very large, return it modulo 109 + 7.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Math · Dynamic Programming

Example 1

3
3
1

Example 2

2
3
0

Example 3

2
3
1

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Largest constraint values

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #920: Number of Music Playlists
class Solution {
    public int numMusicPlaylists(int n, int goal, int k) {
        final int mod = (int) 1e9 + 7;
        long[][] f = new long[goal + 1][n + 1];
        f[0][0] = 1;
        for (int i = 1; i <= goal; ++i) {
            for (int j = 1; j <= n; ++j) {
                f[i][j] = f[i - 1][j - 1] * (n - j + 1);
                if (j > k) {
                    f[i][j] += f[i - 1][j] * (j - k);
                }
                f[i][j] %= mod;
            }
        }
        return (int) f[goal][n];
    }
}

// Accepted solution for LeetCode #920: Number of Music Playlists
func numMusicPlaylists(n int, goal int, k int) int {
	const mod = 1e9 + 7
	f := make([][]int, goal+1)
	for i := range f {
		f[i] = make([]int, n+1)
	}
	f[0][0] = 1
	for i := 1; i <= goal; i++ {
		for j := 1; j <= n; j++ {
			f[i][j] = f[i-1][j-1] * (n - j + 1)
			if j > k {
				f[i][j] += f[i-1][j] * (j - k)
			}
			f[i][j] %= mod
		}
	}
	return f[goal][n]
}

# Accepted solution for LeetCode #920: Number of Music Playlists
class Solution:
    def numMusicPlaylists(self, n: int, goal: int, k: int) -> int:
        mod = 10**9 + 7
        f = [[0] * (n + 1) for _ in range(goal + 1)]
        f[0][0] = 1
        for i in range(1, goal + 1):
            for j in range(1, n + 1):
                f[i][j] = f[i - 1][j - 1] * (n - j + 1)
                if j > k:
                    f[i][j] += f[i - 1][j] * (j - k)
                f[i][j] %= mod
        return f[goal][n]

// Accepted solution for LeetCode #920: Number of Music Playlists
impl Solution {
    #[allow(dead_code)]
    pub fn num_music_playlists(n: i32, goal: i32, k: i32) -> i32 {
        let mut dp: Vec<Vec<i64>> = vec![vec![0; n as usize + 1]; goal as usize + 1];

        // Initialize the dp vector
        dp[0][0] = 1;

        // Begin the dp process
        for i in 1..=goal as usize {
            for j in 1..=n as usize {
                // Choose the song that has not been chosen before
                // We have n - (j - 1) songs to choose
                dp[i][j] += dp[i - 1][j - 1] * ((n - ((j as i32) - 1)) as i64);

                // Choose the song that has been chosen before
                // We have j - k songs to choose if j > k
                if (j as i32) > k {
                    dp[i][j] += dp[i - 1][j] * (((j as i32) - k) as i64);
                }

                // Update dp[i][j]
                dp[i][j] %= ((1e9 as i32) + 7) as i64;
            }
        }

        dp[goal as usize][n as usize] as i32
    }
}

// Accepted solution for LeetCode #920: Number of Music Playlists
function numMusicPlaylists(n: number, goal: number, k: number): number {
    const mod = 1e9 + 7;
    const f = new Array(goal + 1).fill(0).map(() => new Array(n + 1).fill(0));
    f[0][0] = 1;
    for (let i = 1; i <= goal; ++i) {
        for (let j = 1; j <= n; ++j) {
            f[i][j] = f[i - 1][j - 1] * (n - j + 1);
            if (j > k) {
                f[i][j] += f[i - 1][j] * (j - k);
            }
            f[i][j] %= mod;
        }
    }
    return f[goal][n];
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(goal × n)

Space

O(goal × n)

Approach Breakdown

RECURSIVE

O(2ⁿ) time

O(n) space

Pure recursion explores every possible choice at each step. With two choices per state (take or skip), the decision tree has 2ⁿ leaves. The recursion stack uses O(n) space. Many subproblems are recomputed exponentially many times.

DYNAMIC PROGRAMMING

O(n × m) time

O(n × m) space

Each cell in the DP table is computed exactly once from previously solved subproblems. The table dimensions determine both time and space. Look for the state variables — each unique combination of state values is one cell. Often a rolling array can reduce space by one dimension.

Shortcut: Count your DP state dimensions → that’s your time. Can you drop one? That’s your space optimization.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.

State misses one required dimension

Wrong move: An incomplete state merges distinct subproblems and caches incorrect answers.

Usually fails on: Correctness breaks on cases that differ only in hidden state.

Fix: Define state so each unique subproblem maps to one DP cell.