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Formatted question description: https://leetcode.ca/all/1866.html

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

Hard

## 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 10^9 + 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

## Solution

Use dynamic programming. Create a 2D array dp with n + 1 rows and n + 1 columns, where dp[i][j] represents the number of ways to rearrange i sticks with j sticks visible, where i >= j >= 1.

The base case of dp is that dp[i][i] = 1 for all 1 <= i <= n. For the rest values, dp[i][j] = dp[i - 1][j - 1] + dp[i - 1][j] * (i - 1).

Finally, return dp[n][k].

• class Solution {
public int rearrangeSticks(int n, int k) {
final int MODULO = 1000000007;
int[][] dp = new int[n + 1][n + 1];
for (int i = 1; i <= n; i++)
dp[i][i] = 1;
for (int i = 2; i <= n; i++) {
dp[i][1] = (int) ((long) dp[i - 1][1] * (i - 1) % MODULO);
for (int j = 2; j <= i; j++)
dp[i][j] = (int) ((dp[i - 1][j - 1] + (long) dp[i - 1][j] * (i - 1)) % MODULO);
}
return dp[n][k];
}
}

############

class Solution {
public int rearrangeSticks(int n, int k) {
final int mod = (int) 1e9 + 7;
int[] f = new int[k + 1];
f[0] = 1;
for (int i = 1; i <= n; ++i) {
for (int j = k; j > 0; --j) {
f[j] = (int) ((f[j] * (i - 1L) + f[j - 1]) % mod);
}
f[0] = 0;
}
return f[k];
}
}

• // OJ: https://leetcode.com/problems/number-of-ways-to-rearrange-sticks-with-k-sticks-visible/
// Time: O(NK)
// Space: O(Nk)
long dp[1001][1001] = {}, fact[1001] = {}, mod = 1e9 + 7;
class Solution {
long dfs(int n, int k) {
if (n < k) return 0;
if (k == n) return 1;
if (k == 1) return fact[n - 1];
if (dp[n][k]) return dp[n][k];
long ans = 0;
ans = (ans + dfs(n - 1, k) * (n - 1) % mod) % mod;
ans = (ans + dfs(n - 1, k - 1)) % mod;
return dp[n][k] = ans;
}
public:
int rearrangeSticks(int n, int k) {
memset(dp, 0, sizeof(dp));
fact[0] = 1;
for (int i = 1; i <= n; ++i) {
fact[i] = (fact[i - 1] * i) % mod; // fact(n) = n!
}
return dfs(n, k);
}
};

• # 1866. Number of Ways to Rearrange Sticks With K Sticks Visible
# https://leetcode.com/problems/number-of-ways-to-rearrange-sticks-with-k-sticks-visible

class Solution:
@functools.lru_cache(None)
def rearrangeSticks(self, n, k, M = 10**9 + 7):
if n == k: return 1
if k == 0: return 0
return (self.rearrangeSticks(n - 1, k - 1) + self.rearrangeSticks(n - 1, k) * (n - 1)) % M


• func rearrangeSticks(n int, k int) int {
const mod = 1e9 + 7
f := make([]int, k+1)
f[0] = 1
for i := 1; i <= n; i++ {
for j := k; j > 0; j-- {
f[j] = (f[j-1] + f[j]*(i-1)) % mod
}
f[0] = 0
}
return f[k]
}

• function rearrangeSticks(n: number, k: number): number {
const mod = 10 ** 9 + 7;
const f: number[] = Array(n + 1).fill(0);
f[0] = 1;
for (let i = 1; i <= n; ++i) {
for (let j = k; j; --j) {
f[j] = (f[j] * (i - 1) + f[j - 1]) % mod;
}
f[0] = 0;
}
return f[k];
}