1866. Number of Ways to Rearrange Sticks With K Sticks Visible

Description

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

 

Solutions

Solution: Dynamic Programming

• Time complexity: O(nk)
• Space complexity: O(nk)

 

JavaScript

/**
 * @param {number} n
 * @param {number} k
 * @return {number}
 */
const rearrangeSticks = function (n, k) {
  const MODULO = BigInt(10 ** 9 + 7);
  const dp = Array.from({ length: n + 1 }, () => new Array(k + 1).fill(-1));

  const arrangeSticks = (sticks, visible) => {
    if (!sticks && !visible) return 1n;
    if (!sticks || !visible || visible > sticks) return 0n;
    if (dp[sticks][visible] !== -1) return dp[sticks][visible];
    const visibleArrange = arrangeSticks(sticks - 1, visible - 1);
    const unVisibleArrange = BigInt(sticks - 1) * arrangeSticks(sticks - 1, visible);
    const result = (visibleArrange + unVisibleArrange) % MODULO;

    dp[sticks][visible] = result;

    return result;
  };

  return Number(arrangeSticks(n, k));
};