Ira and Flamenco
Description
Ira loves Spanish flamenco dance very much. She decided to start her own dance studio and found $n$ students, $i$th of whom has level $a_i$.
Ira can choose several of her students and set a dance with them. So she can set a huge number of dances, but she is only interested in magnificent dances. The dance is called magnificent if the following is true:
- exactly $m$ students participate in the dance;
- levels of all dancers are pairwise distinct;
- levels of every two dancers have an absolute difference strictly less than $m$.
For example, if $m = 3$ and $a = [4, 2, 2, 3, 6]$, the following dances are magnificent (students participating in the dance are highlighted in red): $[\color{red}{4}, 2, \color{red}{2}, \color{red}{3}, 6]$, $[\color{red}{4}, \color{red}{2}, 2, \color{red}{3}, 6]$. At the same time dances $[\color{red}{4}, 2, 2, \color{red}{3}, 6]$, $[4, \color{red}{2}, \color{red}{2}, \color{red}{3}, 6]$, $[\color{red}{4}, 2, 2, \color{red}{3}, \color{red}{6}]$ are not magnificent.
In the dance $[\color{red}{4}, 2, 2, \color{red}{3}, 6]$ only $2$ students participate, although $m = 3$.
The dance $[4, \color{red}{2}, \color{red}{2}, \color{red}{3}, 6]$ involves students with levels $2$ and $2$, although levels of all dancers must be pairwise distinct.
In the dance $[\color{red}{4}, 2, 2, \color{red}{3}, \color{red}{6}]$ students with levels $3$ and $6$ participate, but $|3 - 6| = 3$, although $m = 3$.
Help Ira count the number of magnificent dances that she can set. Since this number can be very large, count it modulo $10^9 + 7$. Two dances are considered different if the sets of students participating in them are different.
The first line contains a single integer $t$ ($1 \le t \le 10^4$) — number of testcases.
The first line of each testcase contains integers $n$ and $m$ ($1 \le m \le n \le 2 \cdot 10^5$) — the number of Ira students and the number of dancers in the magnificent dance.
The second line of each testcase contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — levels of students.
It is guaranteed that the sum of $n$ over all testcases does not exceed $2 \cdot 10^5$.
For each testcase, print a single integer — the number of magnificent dances. Since this number can be very large, print it modulo $10^9 + 7$.
Input
The first line contains a single integer $t$ ($1 \le t \le 10^4$) — number of testcases.
The first line of each testcase contains integers $n$ and $m$ ($1 \le m \le n \le 2 \cdot 10^5$) — the number of Ira students and the number of dancers in the magnificent dance.
The second line of each testcase contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — levels of students.
It is guaranteed that the sum of $n$ over all testcases does not exceed $2 \cdot 10^5$.
Output
For each testcase, print a single integer — the number of magnificent dances. Since this number can be very large, print it modulo $10^9 + 7$.
9
7 4
8 10 10 9 6 11 7
5 3
4 2 2 3 6
8 2
1 5 2 2 3 1 3 3
3 3
3 3 3
5 1
3 4 3 10 7
12 3
5 2 1 1 4 3 5 5 5 2 7 5
1 1
1
3 2
1 2 3
2 2
1 2
5
2
10
0
5
11
1
2
1
Note
In the first testcase, Ira can set such magnificent dances: $[\color{red}{8}, 10, 10, \color{red}{9}, \color{red}{6}, 11, \color{red}{7}]$, $[\color{red}{8}, \color{red}{10}, 10, \color{red}{9}, 6, 11, \color{red}{7}]$, $[\color{red}{8}, 10, \color{red}{10}, \color{red}{9}, 6, 11, \color{red}{7}]$, $[\color{red}{8}, 10, \color{red}{10}, \color{red}{9}, 6, \color{red}{11}, 7]$, $[\color{red}{8}, \color{red}{10}, 10, \color{red}{9}, 6, \color{red}{11}, 7]$.
The second testcase is explained in the statements.