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_pattern_{1} ' ' card_pattern_{2} ' ' ... ' ' card_pattern_{L}
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.
The input is a sequence of datasets. Each dataset is formatted as follows.
N M L
card_pattern_{1} card_pattern_{2} ... card_pattern_{L}
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
L ≤ N × 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.
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.
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