A12141. Palindrome Pairs

After learning a lot about space exploration, a little girl named Ana wants to change the subject.

Ana is a girl who loves palindromes (string that can be read the same backwards as forward). She has learned how to check for a given string whether it's a palindrome or not, but soon she grew tired of this problem, so she came up with a more interesting one and she needs your help to solve it:

You are given an array of strings which consist of only small letters of the alphabet. Your task is to find how many palindrome pairs are there in the array. A palindrome pair is a pair of strings such that the following condition holds: at least one permutation of the concatenation of the two strings is a palindrome. In other words, if you have two strings, let's say "aab" and "abcac", and you concatenate them into "aababcac", we have to check if there exists a permutation of this new string such that it is a palindrome (in this case there exists the permutation "aabccbaa").

Two pairs are considered different if the strings are located on different indices. The pair of strings with indices $(i,j)$ is considered the same as the pair $(j,i)$ .

The first line contains a positive integer $N$ ( $1 \le N \le 100\,000$ ), representing the length of the input array.

Eacg of the next $N$ lines contains a string (consisting of lowercase English letters from 'a' to 'z') — an element of the input array.

The total number of characters in the input array will be less than $1\,000\,000$ .

Output one number, representing how many palindrome pairs there are in the array.

The first example:

1. aa $+$ bb $\to$ abba.

The second example:

1. aab $+$ abcac $=$ aababcac $\to$ aabccbaa
2. aab $+$ aa $=$ aabaa
3. abcac $+$ aa $=$ abcacaa $\to$ aacbcaa
4. dffe $+$ ed $=$ dffeed $\to$ fdeedf
5. dffe $+$ aade $=$ dffeaade $\to$ adfaafde
6. ed $+$ aade $=$ edaade $\to$ aeddea