You are given three integers n, m, and k. Consider the following algorithm used to find the maximum element in an array of positive integers:

• Initialize maximum_value = -1 and search_cost = 0.

• Iterate through each element x in the array arr:

  • If x is strictly greater than maximum_value, set maximum_value = x and increase search_cost by $1$.

• Return search_cost.

You need to construct an array arr of length n such that:

• Each element arr[i] satisfies $1 \leq$ arr[i] $\leq$ m for all $0 \leq$ i $<$ n.

• Applying the algorithm on arr results in a search_cost equal to k.

Return the total number of valid arrays that can be constructed. Since the answer may be very large, return it modulo $10^9 + 7$.

Constraints:

• $1 \leq$ n $\leq 50$

• $1 \leq$ m $\leq 100$

• $0 \leq$ k $\leq$ n

Now, let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

Implement your solution in the following coding playground.