AND Sequences

2000ms

262144K

Description:

A sequence of $$$n$$$ non-negative integers ($$$n \ge 2$$$) $$$a_1, a_2, \dots, a_n$$$ is called good if for all $$$i$$$ from $$$1$$$ to $$$n-1$$$ the following condition holds true: $$$$$$a_1 \: \& \: a_2 \: \& \: \dots \: \& \: a_i = a_{i+1} \: \& \: a_{i+2} \: \& \: \dots \: \& \: a_n,$$$$$$ where $$$\&$$$ denotes the bitwise AND operation.

You are given an array $$$a$$$ of size $$$n$$$ ($$$n \geq 2$$$). Find the number of permutations $$$p$$$ of numbers ranging from $$$1$$$ to $$$n$$$, for which the sequence $$$a_{p_1}$$$, $$$a_{p_2}$$$, ... ,$$$a_{p_n}$$$ is good. Since this number can be large, output it modulo $$$10^9+7$$$.

Input:

The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$), denoting the number of test cases.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 2 \cdot 10^5$$$) — the size of the array.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \le a_i \le 10^9$$$) — the elements of the array.

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

Output:

Output $$$t$$$ lines, where the $$$i$$$-th line contains the number of good permutations in the $$$i$$$-th test case modulo $$$10^9 + 7$$$.

Sample Input:

Sample Output:

Note:

In the first test case, since all the numbers are equal, whatever permutation we take, the sequence is good. There are a total of $$$6$$$ permutations possible with numbers from $$$1$$$ to $$$3$$$: $$$[1,2,3]$$$, $$$[1,3,2]$$$, $$$[2,1,3]$$$, $$$[2,3,1]$$$, $$$[3,1,2]$$$, $$$[3,2,1]$$$.

In the second test case, it can be proved that no permutation exists for which the sequence is good.

In the third test case, there are a total of $$$36$$$ permutations for which the sequence is good. One of them is the permutation $$$[1,5,4,2,3]$$$ which results in the sequence $$$s=[0,0,3,2,0]$$$. This is a good sequence because

$$$ s_1 = s_2 \: \& \: s_3 \: \& \: s_4 \: \& \: s_5 = 0$$$,
$$$ s_1 \: \& \: s_2 = s_3 \: \& \: s_4 \: \& \: s_5 = 0$$$,
$$$ s_1 \: \& \: s_2 \: \& \: s_3 = s_4 \: \& \: s_5 = 0$$$,
$$$ s_1 \: \& \: s_2 \: \& \: s_3 \: \& \: s_4 = s_5 = 0$$$.

Informação

Provedor Codeforces

Código CF1513B