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| #include<iostream>
#include<cstdio>
#include<cstring>
#include<string>
#include<algorithm>
#include<cassert>
#include<vector>
using namespace std;
#define MAXN 1002
#define pb push_back
int att[MAXN],n,m,p,T;
typedef long long ll;
struct frac
{
ll up, low;
frac(ll up = 0, ll low = 1)
{
if (low < 0) up = -up, low = -low;
assert(low);
ll g = __gcd(abs(up), low);
this->up = up / g, this->low = low / g;
}
void output()
{
ll g = __gcd(abs(up), low);
this->up /= g;
this->low /= g;
if(up==0) low=1;
printf("%lld/%lld\n", up, low);
}
frac operator + (const frac &b) const
{
return frac(up * b.low + low * b.up, low * b.low);
}
frac operator - (const frac &b) const
{
return frac(up * b.low - low * b.up, low * b.low);
}
frac operator * (const frac &b) const
{
return frac(up * b.up, low * b.low);
}
frac operator / (const frac &b) const
{
return frac(up * b.low, low * b.up);
}
bool operator < (const frac &b) const
{
return up * b.low < low * b.up;
}
bool operator == (const frac &b) const
{
return up * b.low == low * b.up;
}
bool operator > (const frac& b) const
{
return b < *this;
}
bool operator <= (const frac& b) const
{
return !(b < *this);
}
bool operator >= (const frac &b) const
{
return !(*this < b);
}
bool operator != (const frac &b) const
{
return up * b.low != low * b.up;
}
frac operator += (const frac &b)
{
return *this = *this + b;
}
frac operator -= (const frac &b)
{
return*this = *this - b;
}
frac operator *= (const frac &b)
{
return *this = *this * b;
}
frac operator /= (const frac &b)
{
return *this = *this / b;
}
};
vector <frac> dp[22];
frac dfs(int n,int st,int p,int left)
{
if(n<0) return frac(0,1);
frac &res=dp[n][st];
if(res!=frac(-1,1)) return res;
res=frac(0,1);
if(p<=0) return res=frac(1,1);
if(left<=0)
return res=frac(0,1);
int cm=0;
for(int i=1;i<=m;i++){
int mm=(1<<(i-1));
if((st&mm)==0) cm++;
}
if(left>=n+cm){
int tot=0;
for(int i=1;i<=m;i++){
int mm=(1<<(i-1));
if((st&(1<<(i-1)))==0) tot+=att[i];
}
if(p<=tot) return res=frac(1,1);
else return res=frac(0,1);
}
res+=dfs(n-1,st,p,left+1)*frac(n,n+cm);
for(int i=1;i<=m;i++){
int mm=(1<<(i-1));
if((st&mm)==0){
res+=(dfs(n,st|mm,p-att[i],left-1)*frac(1,n+cm));
}
}
return res;
}
void init()
{
int lim=((1<<(m)));
for(int i=0;i<=n;i++){
dp[i].resize(lim+1);
for(int k=0;k<=lim;k++)
dp[i][k]=frac(-1,1);
}
}
int main()
{
scanf("%d",&T);
while(T--){
scanf("%d%d%d",&p,&n,&m);
init();
for(int i=1;i<=m;i++)
scanf("%d",&att[i]);
frac ans=dfs(n,0,p,1);
ans.output();
}
return 0;
}
|