Score : $1200$ points

Problem Statement

You are given a simple connected undirected graph $G$ consisting of $N$ vertices and $M$ edges. The vertices are numbered $1$ to $N$, and the edges are numbered $1$ to $M$.

Edge $i$ connects Vertex $a_i$ and $b_i$ bidirectionally. It is guaranteed that the subgraph consisting of Vertex $1,2,\ldots,N$ and Edge $1,2,\ldots,N-1$ is a spanning tree of $G$.

An allocation of weights to the edges is called a good allocation when the tree consisting of Vertex $1,2,\ldots,N$ and Edge $1,2,\ldots,N-1$ is a minimum spanning tree of $G$.

There are $M!$ ways to allocate the edges distinct integer weights between $1$ and $M$. For each good allocation among those, find the total weight of the edges in the minimum spanning tree, and print the sum of those total weights modulo $10^{9}+7$.

Constraints

• All values in input are integers.
• $2 \leq N \leq 20$
• $N-1 \leq M \leq N(N-1)/2$
• $1 \leq a_i, b_i \leq N$
• $G$ does not have self-loops or multiple edges.
• The subgraph consisting of Vertex $1,2,\ldots,N$ and Edge $1,2,\ldots,N-1$ is a spanning tree of $G$.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$a_1$ $b_1$

$\vdots$

$a_M$ $b_M$

Output

Print the answer.

3 3
1 2
2 3
1 3

6

An allocation is good only if Edge $3$ has the weight $3$. For these good allocations, the total weight of the edges in the minimum spanning tree is $3$, and there are two good allocations, so the answer is $6$.

4 4
1 2
3 2
3 4
1 3

50

15 28
10 7
5 9
2 13
2 14
6 1
5 12
2 10
3 9
10 15
11 12
12 6
2 12
12 8
4 10
15 3
13 14
1 15
15 12
4 14
1 7
5 11
7 13
9 10
2 7
1 9
5 6
12 14
5 2

657573092

Print the sum of those total weights modulo $10^{9}+7$.