Eleventh Birthday

题意翻译

本题是多组数据 对于每组数据，给出n个数字，求有多少种排列方式，使得排列后的n个数字首尾相接形成的数字能被11整除。答案对998244353取模 感谢@ObsdianGungnir 提供的翻译。

题目描述

It is Borya's eleventh birthday, and he has got a great present: $ n $ cards with numbers. The $ i $ -th card has the number $ a_{i} $ written on it. Borya wants to put his cards in a row to get one greater number. For example, if Borya has cards with numbers $ 1 $ , $ 31 $ , and $ 12 $ , and he puts them in a row in this order, he would get a number $ 13112 $ . He is only 11, but he already knows that there are $ n! $ ways to put his cards in a row. But today is a special day, so he is only interested in such ways that the resulting big number is divisible by eleven. So, the way from the previous paragraph is good, because $ 13112=1192×11 $ , but if he puts the cards in the following order: $ 31 $ , $ 1 $ , $ 12 $ , he would get a number $ 31112 $ , it is not divisible by $ 11 $ , so this way is not good for Borya. Help Borya to find out how many good ways to put the cards are there. Borya considers all cards different, even if some of them contain the same number. For example, if Borya has two cards with 1 on it, there are two good ways. Help Borya, find the number of good ways to put the cards. This number can be large, so output it modulo $ 998244353 $ .

输入输出格式

输入格式

Input data contains multiple test cases. The first line of the input data contains an integer $ t $ — the number of test cases ( $ 1<=t<=100 $ ). The descriptions of test cases follow. Each test is described by two lines. The first line contains an integer $ n $ ( $ 1<=n<=2000 $ ) — the number of cards in Borya's present. The second line contains $ n $ integers $ a_{i} $ ( $ 1<=a_{i}<=10^{9} $ ) — numbers written on the cards. It is guaranteed that the total number of cards in all tests of one input data doesn't exceed $ 2000 $ .

输出格式

For each test case output one line: the number of ways to put the cards to the table so that the resulting big number was divisible by 11, print the number modulo $ 998244353 $ .

输入输出样例

输入样例 #1

4
2
1 1
3
1 31 12
3
12345 67 84
9
1 2 3 4 5 6 7 8 9

输出样例 #1

2
2
2
31680