A15870 | Ira and Flamenco

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): $[\color{red}{4}\color{black}, 2,\color{red}{2}\color{black},\color{red}{3}\color{black}, 6]$ , $[\color{red}{4}\color{black}, \color{red}{2}\color{black}, 2, \color{red}{3}\color{black}, 6]$ . At the same time dances $[\color{red}{4}\color{black}, 2, 2, \color{red}{3}\color{black}, 6]$ , $[4, \color{red}{2}\color{black}, \color{red}{2}\color{black}, \color{red}{3}\color{black}, 6]$ , $[\color{red}{4}\color{black}, 2, 2, \color{red}{3}\color{black}, \color{red}{6}\color{black}]$ are not magnificent.

In the dance $[\color{red}{4}\color{black}, 2, 2, \color{red}{3}\color{black}, 6]$ only $2$ students participate, although $m = 3$ .

The dance $[4, \color{red}{2}\color{black}, \color{red}{2}\color{black}, \color{red}{3}\color{black}, 6]$ involves students with levels $2$ and $2$ , although levels of all dancers must be pairwise distinct.

In the dance $[\color{red}{4}\color{black}, 2, 2, \color{red}{3}\color{black}, \color{red}{6}\color{black}]$ 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 $10^9 + 7$ .