Sonya and Informatics
Description
A girl named Sonya is studying in the scientific lyceum of the Kingdom of Kremland. The teacher of computer science (Sonya's favorite subject!) invented a task for her.
Given an array $a$ of length $n$, consisting only of the numbers $0$ and $1$, and the number $k$. Exactly $k$ times the following happens:
- Two numbers $i$ and $j$ are chosen equiprobable such that ($1 \leq i < j \leq n$).
- The numbers in the $i$ and $j$ positions are swapped.
Sonya's task is to find the probability that after all the operations are completed, the $a$ array will be sorted in non-decreasing order. She turned to you for help. Help Sonya solve this problem.
It can be shown that the desired probability is either $0$ or it can be represented as $\dfrac{P}{Q}$, where $P$ and $Q$ are coprime integers and $Q \not\equiv 0~\pmod {10^9+7}$.
The first line contains two integers $n$ and $k$ ($2 \leq n \leq 100, 1 \leq k \leq 10^9$) — the length of the array $a$ and the number of operations.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 1$) — the description of the array $a$.
If the desired probability is $0$, print $0$, otherwise print the value $P \cdot Q^{-1}$ $\pmod {10^9+7}$, where $P$ and $Q$ are defined above.
Input
The first line contains two integers $n$ and $k$ ($2 \leq n \leq 100, 1 \leq k \leq 10^9$) — the length of the array $a$ and the number of operations.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 1$) — the description of the array $a$.
Output
If the desired probability is $0$, print $0$, otherwise print the value $P \cdot Q^{-1}$ $\pmod {10^9+7}$, where $P$ and $Q$ are defined above.
Samples
3 2
0 1 0
333333336
5 1
1 1 1 0 0
0
6 4
1 0 0 1 1 0
968493834
Note
In the first example, all possible variants of the final array $a$, after applying exactly two operations: $(0, 1, 0)$, $(0, 0, 1)$, $(1, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$, $(0, 0, 1)$, $(1, 0, 0)$, $(0, 1, 0)$. Therefore, the answer is $\dfrac{3}{9}=\dfrac{1}{3}$.
In the second example, the array will not be sorted in non-decreasing order after one operation, therefore the answer is $0$.