count-the-number-of-arrays-with-k-matching-adjacent-elements

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