Codeforces Round #275 (Div. 1)C（状压+期望）

C. Game with Strings

time limit per test

1 second

memory limit per test

256 megabytes

input

standard input

output

standard output

You play the game with your friend. The description of this game is listed below.

Your friend creates n distinct strings of the same length m and tells you all the strings. Then he randomly chooses one of them. He chooses strings equiprobably, i.e. the probability of choosing each of the n strings equals . You want to guess which string was chosen by your friend.

In order to guess what string your friend has chosen, you are allowed to ask him questions. Each question has the following form: «What character stands on position pos in the string you have chosen?» A string is considered guessed when the answers to the given questions uniquely identify the string. After the string is guessed, you stop asking questions.

You do not have a particular strategy, so as each question you equiprobably ask about a position that hasn't been yet mentioned. Your task is to determine the expected number of questions needed to guess the string chosen by your friend.

Input

The first line contains a single integer n (1 ≤ n ≤ 50) — the number of strings your friend came up with.

The next n lines contain the strings that your friend has created. It is guaranteed that all the strings are distinct and only consist of large and small English letters. Besides, the lengths of all strings are the same and are between 1 to 20 inclusive.

Output

Print the single number — the expected value. Your answer will be considered correct if its absolute or relative error doesn't exceed 10 - 9.

Sample test(s)

input

2aabaac

output

2.000000000000000

input

3aaAaBaCaa

output

1.666666666666667

input

3acavacwqq

output

1.000000000000000

Note

In the first sample the strings only differ in the character in the third position. So only the following situations are possible:

• you guess the string in one question. The event's probability is ;
• you guess the string in two questions. The event's probability is  ·  =  (as in this case the first question should ask about the position that is other than the third one);
• you guess the string in three questions. The event's probability is  ·  ·  = ;

Thus, the expected value is equal to 

In the second sample we need at most two questions as any pair of questions uniquely identifies the string. So the expected number of questions is .

In the third sample whatever position we ask about in the first question, we immediately identify the string.

题意：RT

思路：f[s]表示s二进制状态下，s里面1的数量是已经猜过的位置的字符，0表示还没有猜的

            转移情况为：f[s]由任何的0转移过来，这个过程其实是逆向的，从终点推向起点

            设s的下一步状态为p，即p一定比s多一个1，那么f[s]=1/d∑( m/n*1 + f[p] )  其中d为s的下一步的状态数，

            m为当前s状态的合法字符串数，n为总的字符串，f[p]为下一个状态的期望

            最后f[0]就是答案，即起点