Codeforces Round 964 (Div. 4) |
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Finished |
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 $$$10^9 + 7$$$.
84 31 0 0 15 11 1 1 1 15 50 1 0 1 06 31 0 1 0 1 14 31 0 1 15 31 0 1 1 02 10 034 171 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
In the first test case, there are four subsequences of $$$[1,0,0,1]$$$ with length $$$k=3$$$:
In the second test case, all subsequences of length $$$1$$$ have median $$$1$$$, so the answer is $$$5$$$.
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