Generic Poker

Time Limit : 8 sec, Memory Limit : 65536 KB
Japanese version is here

Generic Poker

You have a deck of N × M cards. Each card in the deck has a rank. The range of ranks is 1 through M, and the deck includes N cards of each rank.

We denote a card with rank m by m here.

You can draw a hand of L cards at random from the deck. If the hand matches the given pattern, some bonus will be rewarded. A pattern is described as follows.

hand_pattern = card_pattern1 ' ' card_pattern2 ' ' ... ' ' card_patternL
card_pattern = '*' | var_plus
var_plus = variable | var_plus '+'
variable = 'a' | 'b' | 'c'

hand_pattern
A hand matches the hand_pattern if each card_pattern in the hand_pattern matches with a distinct card in the hand.
card_pattern
If the card_pattern is an asterisk ('*'), it matches any card. Characters 'a', 'b', and 'c' denote variables and all the occurrences of the same variable match cards of the same rank. A card_pattern with a variable followed by plus ('+') characters matches a card whose rank is the sum of the rank corresponding to the variable and the number of plus characters. You can assume that, when a hand_pattern includes a card_pattern with a variable followed by some number of plus characters, it also includes card_patterns with that variable and all smaller numbers (including zero) of plus characters. For example, if 'a+++' appears in a hand_pattern, card_patterns 'a', 'a+', and 'a++' also appear in the hand_pattern.

There is no restriction on which ranks different variables mean. For example, 'a' and 'b' may or may not match cards of the same rank.

We show some example hand_patterns. The pattern

a * b a b 
matches the hand:
3 3 10 10 9
with 'a's and 'b's meaning 3 and 10 (or 10 and 3), respectively. This pattern also matches the following hand.
3 3 3 3 9
In this case, both 'a's and 'b's mean 3. The pattern
a a+ a++ a+++ a++++
matches the following hand.
4 5 6 7 8
In this case, 'a' should mean 4.

Your mission is to write a program that computes the probability that a hand randomly drawn from the deck matches the given hand_pattern.

Input

The input is a sequence of datasets. Each dataset is formatted as follows.

N M L
card_pattern1 card_pattern2 ... card_patternL

The first line consists of three positive integers N, M, and L. N indicates the number of cards in each rank, M indicates the number of ranks, and L indicates the number of cards in a hand. N, M, and L are constrained as follows.

1 ≤ N ≤ 7
1 ≤ M ≤ 60
1 ≤ L ≤ 7
LN × M

The second line describes a hand_pattern.

The end of the input is indicated by a line containing three zeros separated by a single space.

Output

For each dataset, output a line containing a decimal fraction which means the probability of a hand matching the hand_pattern.

The output should not contain an error greater than 10−8.

No other characters should be contained in the output.

Sample Input

1 1 1
a
3 3 4
a+ * a *
2 2 3
a a b
2 2 3
* * *
2 2 3
* b b
2 2 2
a a
2 3 3
a a+ a++
2 6 6
a a+ a++ b b+ b++
4 13 5
a a * * *
4 13 5
a a b b *
4 13 5
a a a * *
4 13 5
a a+ a++ a+++ a++++
4 13 5
* * * * *
4 13 5
a a a b b
4 13 5
a a a a *
7 60 7
a b a b c c *
7 60 7
* * * * * * *
7 60 7
a a+ a++ a+++ a++++ a+++++ a++++++
1 14 4
b a+ a a
0 0 0

Output for the Sample Input

1.0000000000
0.8809523810
1.0000000000
1.0000000000
1.0000000000
0.3333333333
0.4000000000
0.1212121212
0.4929171669
0.0492196879
0.0228091236
0.0035460338
1.0000000000
0.0014405762
0.0002400960
0.0002967709
1.0000000000
0.0000001022
0.0000000000

Source: ACM International Collegiate Programming Contest , Japan Domestic Contest, Tokyo, Japan, 2012-07-6
http://www.cs.titech.ac.jp/icpc2012/