NC222419. [USACOOPEN2021G]Permutation
描述
Bessie has %7D)
favorite distinct points on a 
grid, no three of which are collinear. For each 
, the i-th point is denoted by two integers 
and %7D)
.
Bessie draws some segments between the points as follows.
Bessie draws some segments between the points as follows.
- She chooses some permutation
of the
points.
- She draws segments between
and
and
, and
- Then for each integer ii from 4 to N in order, she draws a line segment from pi to pj for all
such that the segment does not intersect any previously drawn segments (aside from at endpoints).
Bessie notices that for each i, she drew exactly three new segments. Compute the number of permutations Bessie could have chosen on step 1 that would satisfy this property, modulo .
输入描述
The first line contains N.
Followed by N lines, each containing two space-separated integersand
.
输出描述
The number of permutations modulo
示例1
输入:
4 0 0 0 4 1 1 1 2
输出:
0
说明:
No permutations work.示例2
输入:
4 0 0 0 4 4 0 1 1
输出:
24
说明:
All permutations work.示例3
输入:
5 0 0 0 4 4 0 1 1 1 2
输出:
96
说明:
One permutation that satisfies the property is (0,0),(0,4),(4,0),(1,2),(1,1). For this permutation,
First, she draws segments between every pair of (0,0),(0,4), and (4,0).Then she draws segments from (0,0),(0,4), and (4,0) to (1,2).Finally, she draws segments from (1,2), (4,0), and (0,0) to (1,1).Diagram:
