Expected Median
Description
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.
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 $2 \cdot 10^5$.
For each test case, print the sum modulo $10^9 + 7$.
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 $10^9 + 7$.
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
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 $= 0$.
In the second test case, all subsequences of length $1$ have median $1$, so the answer is $5$.