F. Expected Median
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Arul has a binary array$$$^{\text{∗}}$$$ $$$a$$$ of length $$$n$$$.

He will take all subsequences$$$^{\text{†}}$$$ of length $$$k$$$ ($$$k$$$ is odd) of this array and find their median.$$$^{\text{‡}}$$$

What is the sum of all these values?

As this sum can be very large, output it modulo $$$10^9 + 7$$$. In other words, print the remainder of this sum when divided by $$$10^9 + 7$$$.

$$$^{\text{∗}}$$$A binary array is an array consisting only of zeros and ones.

$$$^{\text{†}}$$$An array $$$b$$$ is a subsequence of an array $$$a$$$ if $$$b$$$ can be obtained from $$$a$$$ by the deletion of several (possibly, zero or all) elements. Subsequences don't have to be contiguous.

$$$^{\text{‡}}$$$The median of an array of odd length $$$k$$$ is the $$$\frac{k+1}{2}$$$-th element when sorted.

Input

The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases.

The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \leq k \leq n \leq 2 \cdot 10^5$$$, $$$k$$$ is odd) — the length of the array and the length of the subsequence, respectively.

The second line of each test case contains $$$n$$$ integers $$$a_i$$$ ($$$0 \leq a_i \leq 1$$$) — the elements of the array.

It is guaranteed that sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output

For each test case, print the sum modulo $$$10^9 + 7$$$.

Example
Input
8
4 3
1 0 0 1
5 1
1 1 1 1 1
5 5
0 1 0 1 0
6 3
1 0 1 0 1 1
4 3
1 0 1 1
5 3
1 0 1 1 0
2 1
0 0
34 17
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Output
2
5
0
16
4
7
0
333606206
Note

In the first test case, there are four subsequences of $$$[1,0,0,1]$$$ with length $$$k=3$$$:

  • $$$[1,0,0]$$$: median $$$= 0$$$.
  • $$$[1,0,1]$$$: median $$$= 1$$$.
  • $$$[1,0,1]$$$: median $$$= 1$$$.
  • $$$[0,0,1]$$$: median $$$= 0$$$.
The sum of the results is $$$0+1+1+0=2$$$.

In the second test case, all subsequences of length $$$1$$$ have median $$$1$$$, so the answer is $$$5$$$.