Introduction

In this blog post, we will explore how to solve the LeetCode problem "1866" using AI. LeetCode is a popular platform for preparing for coding interviews, and with the help of AI tools like Chatmagic, we can generate solutions quickly and efficiently - helping you pass the interviews and get the job offer without having to study for months.

Problem Statement

There are n uniquely-sized sticks whose lengths are integers from 1 to n. You want to arrange the sticks such that exactly k sticks are visible from the left. A stick is visible from the left if there are no longer sticks to the left of it. For example, if the sticks are arranged [1,3,2,5,4], then the sticks with lengths 1, 3, and 5 are visible from the left. Given n and k, return the number of such arrangements. Since the answer may be large, return it modulo 109 + 7. Example 1: Input: n = 3, k = 2 Output: 3 Explanation: [1,3,2], [2,3,1], and [2,1,3] are the only arrangements such that exactly 2 sticks are visible. The visible sticks are underlined. Example 2: Input: n = 5, k = 5 Output: 1 Explanation: [1,2,3,4,5] is the only arrangement such that all 5 sticks are visible. The visible sticks are underlined. Example 3: Input: n = 20, k = 11 Output: 647427950 Explanation: There are 647427950 (mod 109 + 7) ways to rearrange the sticks such that exactly 11 sticks are visible. Constraints: 1 <= n <= 1000 1 <= k <= n

Explanation

Here's the solution:

• Dynamic Programming: The core idea is to use dynamic programming to build up the number of arrangements. dp[n][k] will store the number of arrangements of n sticks with k visible sticks.
• Base Cases and Transitions: We initialize dp[i][i] = 1 for all i, since there's only one way to have all sticks visible (ascending order). The transitions are based on where we place the longest stick (n). We can either place it at the beginning (making it visible) or somewhere else (not visible).

• Modulo Arithmetic: We perform all calculations modulo 10^9 + 7 to prevent overflow.

• Time & Space Complexity: O(n*k) time and space complexity, where n is the number of sticks and k is the number of visible sticks.

Code

    def rearrange_sticks(n: int, k: int) -> int:
    """
    Calculates the number of arrangements of n sticks such that exactly k sticks are visible from the left.

    Args:
        n: The number of sticks.
        k: The number of visible sticks.

    Returns:
        The number of arrangements modulo 10^9 + 7.
    """

    MOD = 10**9 + 7
    dp = [[0] * (k + 1) for _ in range(n + 1)]

    dp[0][0] = 1  # Base case: no sticks, no visible sticks, 1 way

    for i in range(1, n + 1):
        for j in range(1, k + 1):
            # Case 1: The longest stick (i) is visible.
            dp[i][j] = (dp[i][j] + dp[i - 1][j - 1]) % MOD

            # Case 2: The longest stick (i) is not visible.
            dp[i][j] = (dp[i][j] + (dp[i - 1][j] * (i - 1)) % MOD) % MOD

    return dp[n][k]