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G - Similar Permutation /

Time Limit: 4 sec / Memory Limit: 1024 MB

### 問題文

(1,2,\ldots,N) の順列を、以下では単に順列と呼びます。

• (A_{i+1}-A_i)(B_{i+1}-B_i)>0

### 制約

• 2\leq N \leq 100
• 0\leq K \leq N-1
• 10^8 \leq P \leq 10^9
• P は素数
• 入力は全て整数である

### 入力

N K P


### 入力例 1

3 1 282282277


### 出力例 1

16


• A=(1,2,3)
• B=(1,3,2)

この例では、(A_2 - A_1)(B_2 -B_1) > 0, (A_3 - A_2)(B_3 -B_2) < 0 であることから、AB の類似度は 1 だとわかります。

### 入力例 2

50 25 998244353


### 出力例 2

131276976


Score : 600 points

### Problem Statement

Below, a permutation of (1,2,\ldots,N) is simply called a permutation.

For two permutations A=(A_1,A_2,\ldots,A_N),B=(B_1,B_2,\ldots,B_N), let us define their similarity as the number of integers i between 1 and N-1 such that:

• (A_{i+1}-A_i)(B_{i+1}-B_i)>0.

Find the number, modulo a prime number P, of pairs of permutations (A,B) whose similarity is K.

### Constraints

• 2\leq N \leq 100
• 0\leq K \leq N-1
• 10^8 \leq P \leq 10^9
• P is a prime number.
• All values in the input are integers.

### Input

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

N K P


### Sample Input 1

3 1 282282277


### Sample Output 1

16


For instance, below is a pair of permutations that satisfies the condition.

• A=(1,2,3)
• B=(1,3,2)

Here, we have (A_2 - A_1)(B_2 -B_1) > 0 and (A_3 - A_2)(B_3 -B_2) < 0, so the similarity of A and B is 1.

### Sample Input 2

50 25 998244353


### Sample Output 2

131276976


Print the number modulo P.