LeetCode-in-Java

3405. Count the Number of Arrays with K Matching Adjacent Elements

Hard

You are given three integers n, m, k. A good array arr of size n is defined as follows:

• Each element in arr is in the inclusive range [1, m].
• Exactly k indices i (where 1 <= i < n) satisfy the condition arr[i - 1] == arr[i].

Return the number of good arrays that can be formed.

Since the answer may be very large, return it modulo 109 + 7.

Example 1:

Input: n = 3, m = 2, k = 1

Output: 4

Explanation:

• There are 4 good arrays. They are [1, 1, 2], [1, 2, 2], [2, 1, 1] and [2, 2, 1].
• Hence, the answer is 4.

Example 2:

Input: n = 4, m = 2, k = 2

Output: 6

Explanation:

• The good arrays are [1, 1, 1, 2], [1, 1, 2, 2], [1, 2, 2, 2], [2, 1, 1, 1], [2, 2, 1, 1] and [2, 2, 2, 1].
• Hence, the answer is 6.

Example 3:

Input: n = 5, m = 2, k = 0

Output: 2

Explanation:

• The good arrays are [1, 2, 1, 2, 1] and [2, 1, 2, 1, 2]. Hence, the answer is 2.

Constraints:

• 1 <= n <= 105
• 1 <= m <= 105
• 0 <= k <= n - 1

Solution

public class Solution {
    private static final int MOD = (int) (1e9 + 7);

    public int countGoodArrays(int n, int m, int k) {
        long[] f = new long[n + 1];
        f[0] = 1;
        f[1] = 1;
        for (int i = 2; i < f.length; i++) {
            f[i] = (f[i - 1] * i % MOD);
        }
        long ans = comb(n - 1, k, f);
        ans = ans * m % MOD;
        ans = ans * ex(m - 1L, n - k - 1L) % MOD;
        return (int) ans;
    }

    private long ex(long b, long e) {
        long ans = 1;
        while (e > 0) {
            if (e % 2 == 1) {
                ans = (ans * b) % MOD;
            }
            b = (b * b) % MOD;
            e = e >> 1;
        }
        return ans;
    }

    private long comb(int n, int r, long[] f) {
        return f[n] * (ff(f[r])) % MOD * ff(f[n - r]) % MOD;
    }

    private long ff(long x) {
        return ex(x, MOD - 2L);
    }
}

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