Bags with Balls

3000ms

524288K

Description:

There are $$$n$$$ bags, each bag contains $$$m$$$ balls with numbers from $$$1$$$ to $$$m$$$. For every $$$i \in [1, m]$$$, there is exactly one ball with number $$$i$$$ in each bag.

You have to take exactly one ball from each bag (all bags are different, so, for example, taking the ball $$$1$$$ from the first bag and the ball $$$2$$$ from the second bag is not the same as taking the ball $$$2$$$ from the first bag and the ball $$$1$$$ from the second bag). After that, you calculate the number of balls with odd numbers among the ones you have taken. Let the number of these balls be $$$F$$$.

Your task is to calculate the sum of $$$F^k$$$ over all possible ways to take $$$n$$$ balls, one from each bag.

Input:

The first line contains one integer $$$t$$$ ($$$1 \le t \le 5000$$$) — the number of test cases.

Each test case consists of one line containing three integers $$$n$$$, $$$m$$$ and $$$k$$$ ($$$1 \le n, m \le 998244352$$$; $$$1 \le k \le 2000$$$).

Output:

For each test case, print one integer — the sum of $$$F^k$$$ over all possible ways to take $$$n$$$ balls, one from each bag. Since it can be huge, print it modulo $$$998244353$$$.

Sample Input:

5
2 3 8
1 1 1
1 5 10
3 7 2000
1337666 42424242 2000

Sample Output:

Informação

Provedor Codeforces

Código CF1716F