Score : $2000$ points

Problem Statement

Given are integers $N$ and $K$, and a prime number $P$. Find the number, modulo $P$, of directed graphs $G$ with $N$ vertices that satisfy below. Here, the vertices are distinguishable from each other.

• $G$ is a tournament, that is, $G$ contains no duplicated edges or self-loops, and exactly one of the edges $u\to v$ and $v\to u$ exists for any two vertices $u$ and $v$.
• The in-degree of every vertex in $G$ is at most $K$.
• For any four distinct vertices $a$, $b$, $c$, and $d$ in $G$, it is not the case that all of the six edges $a\to b$, $b\to c$, $c\to a$, $a\to d$, $b\to d$, and $c\to d$ exist simultaneously.

Constraints

• $4 \leq N \leq 200$
• $\frac{N-1}{2} \leq K \leq N-1$
• $10^8
• $N$ and $K$ are integers.
• $P$ is a prime number.

Input

Input is given from Standard Input in the following format:

$N$ $K$ $P$

Output

Print the number of directed graphs that satisfy the conditions, modulo $P$.

4 3 998244353

56

Among the $64$ graphs with four vertices, $8$ are isomorphic to the forbidden induced subgraphs, and the other $56$ satisfy the conditions.

7 3 998244353

720

50 37 998244353

495799508