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.</p>

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.
  </li>
• 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.
  </ul> </li> </ul>

  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.

  입력 형식

  The input is a sequence of datasets. Each dataset is formatted as follows.</p>

  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
  • 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.

  출력 형식

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

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

  No other characters should be contained in the output.

  예제 입력

  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

  예제 출력

  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