Problem: 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
>  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.

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






Example

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

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

Solution (in CPP):

Solution:
#include <iostream>
#include <fstream>
#include <algorithm>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cmath>
#include <complex>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <ctime>
#include <deque>
#include <iomanip>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <stack>
#include <sstream>
#include <string>
#include <vector>
using namespace std;

#pragma comment(linker, "/STACK:400000000")

#define EPS 1e-9
#define INF MOD
#define MOD 1000000007LL
#define fir first
#define foreach(it,X) for(it=X.begin();it!=X.end();it++)
#define iss istringstream
#define ite iterator
#define ll long long
#define mp make_pair
#define rep(i,n) rep2(i,0,n)
#define rep2(i,m,n) for(int i=m;i<n;i++)
#define pi pair<int,int>
#define pb push_back
#define sec second
#define sh(i) (1LL<<i)
#define sst stringstream
#define sz size()
#define vi vector<int>
#define vc vector
#define vl vector<ll>
#define vs vector<string>

int N,M,L,a[8],b[8],c[8],x[8];
ll ans;

ll nCk(int n,int k){
	if(n<k)return 0;
	ll res=1;
	rep2(i,n-k+1,n+1)res*=i;
	rep2(i,1,k+1)res/=i;
	return res;
}

void f(int cur,int blank){
	if(x[cur-1]>M)return;
	if(cur==L){
		int count[21]={};
		rep(i,cur)count[x[i]]++;
		rep(i,cur)rep(j,cur)rep(k,cur){
			int needed[21]={};
			rep(l,7){
				needed[x[i]+l]+=a[l];
				needed[x[j]+l]+=b[l];
				needed[x[k]+l]+=c[l];
			}
			ll res=1;
			rep(l,21)if(needed[l]>count[l])goto fail;
			rep(l,14)res*=nCk(N,count[l]);
			ans+=res*nCk(M-x[cur-1]+blank,blank);
			return;
			fail:;
		}
		return;
	}
	x[cur]=x[cur-1];
	f(cur+1,blank);
	x[cur]=x[cur-1]+1;
	f(cur+1,blank);
	x[cur]=x[cur-1]+2;
	f(cur+1,blank+1);
}

int main(){
	while(cin>>N>>M>>L && N){
		rep(i,7)a[i]=b[i]=c[i]=0;
		string h;
		rep(i,L){
			cin>>h;
			if(h[0]=='a')a[h.sz-1]++;
			if(h[0]=='b')b[h.sz-1]++;
			if(h[0]=='c')c[h.sz-1]++;
		}
		ans=0;
		x[0]=1;
		f(1,1);
		printf("%.11f\n",1.*ans/nCk(N*M,L));
	}
}