A15339 | Bags with Balls

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.

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 $k$ ( $1 \le n, m \le 998244352$ ; $1 \le k \le 2000$ ).

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 $998244353$ .