CF1838E.Count Supersequences

You are given an array $a$ of $n$ integers, where all elements $a_i$ lie in the range $[1, k]$ . How many different arrays $b$ of $m$ integers, where all elements $b_i$ lie in the range $[1, k]$ , contain $a$ as a subsequence? Two arrays are considered different if they differ in at least one position.

A sequence $x$ is a subsequence of a sequence $y$ if $x$ can be obtained from $y$ by the deletion of several (possibly, zero or all) elements.

Since the answer may be large, print it modulo $10^9 + 7$ .

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

The first line of each test case contains three integers $n$ , $m$ , $k$ ( $1 \le n \le 2\cdot 10^5$ , $n \le m \le 10^9$ , $1 \le k \le 10^9$ ) — the size of $a$ , the size of $b$ , and the maximum value allowed in the arrays, respectively.

The next line of each test case contains $n$ integers $a_1, a_2, \ldots a_n$ ( $1\le a_i \le k$ ) — the elements of the array $a$ .

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

For each test case, output a single integer — the number of suitable arrays $b$ , modulo $10^9+7$ .

For the first example, since $k=1$ , there is only one array of size $m$ consisting of the integers $[1, k]$ . This array ( $[1, 1, \ldots, 1]$ ) contains the original array as a subsequence, so the answer is 1.

For the second example, the $9$ arrays are $[1, 1, 2, 2]$ , $[1, 2, 1, 2]$ , $[1, 2, 2, 1]$ , $[1, 2, 2, 2]$ , $[1, 2, 2, 3]$ , $[1, 2, 3, 2]$ , $[1, 3, 2, 2]$ , $[2, 1, 2, 2]$ , $[3, 1, 2, 2]$ .

For the fourth example, since $m=n$ , the only array of size $m$ that contains $a$ as a subsequence is $a$ itself.