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Ex - K-Coloring

Score : $600$ points

Problem Statement

You are given a simple undirected graph with $N$ vertices numbered $1$ to $N$ and $M$ edges numbered $1$ to $M$. Edge $i$ connects vertex $u_i$ and vertex $v_i$.

Find the number, modulo $998244353$, of ways to write an integer between $1$ and $K$, inclusive, on each vertex of this graph to satisfy the following condition:

• two vertices connected by an edge always have different numbers written on them.

Constraints

• $1 \leq N \leq 30$
• $0 \leq M \leq \min \left(30, \frac{N(N-1)}{2} \right)$
• $1 \leq K \leq 10^9$
• $1 \leq u_i \lt v_i \leq N$
• The given graph is simple.

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Input

The input is given from Standard Input in the following format:

$N$ $M$ $K$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_M$ $v_M$

Output

Print the number, modulo $998244353$, of ways to write integers between $1$ and $K$, inclusive, on the vertices to satisfy the condition.

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4 3 2
1 2
2 4
2 3

2

Here are the two ways to satisfy the condition.

• Write $1$ on vertices $1, 3, 4$, and write $2$ on vertex $2$.
• Write $1$ on vertex $2$, and write $2$ on vertex $1, 3, 4$.

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4 0 10

10000

All $10^4$ ways satisfy the condition.

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5 10 5
3 5
1 3
1 2
1 4
3 4
2 5
4 5
1 5
2 3
2 4

120

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5 6 294
1 2
2 4
1 3
2 3
4 5
3 5

838338733

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7 12 1000000000
4 5
2 7
3 4
6 7
3 5
5 6
5 7
1 3
4 7
1 5
2 3
3 6

418104233