Problem:
Score : 800 points

Problem Statement

Given are N pairwise distinct non-negative integers A_1,A_2,\ldots,A_N.
Find the number of ways to choose a set of between 1 and K numbers (inclusive) from the given numbers so that the following two conditions are satisfied:

The bitwise AND of the chosen numbers is S.

The bitwise OR of the chosen numbers is T.

Constraints

1 \leq N \leq 50

1 \leq K \leq N

0 \leq A_i < 2^{18}

0 \leq S < 2^{18}

0 \leq T < 2^{18}

A_i \neq A_j (1 \leq i < j \leq N)

Input

Input is given from Standard Input in the following format:

N K S T
A_1 A_2 ... A_N

Output

Print the answer.

Sample Input 1

3 3 0 3
1 2 3

Sample Output 1

2

The conditions are satisfied when we choose \{1,2\} or \{1,2,3\}.

Sample Input 2

5 3 1 7
3 4 9 1 5

Sample Output 2

2

Sample Input 3

5 4 0 15
3 4 9 1 5

Sample Output 3

3

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

   #include <bits/stdc++.h>
//    #include <boost/multiprecision/cpp_int.hpp>
  //  using namespace boost::multiprecision;
 #define int long long
 #define inf  1000000007
 #define pa pair<int,int>
 #define ll long long
 #define pal pair<double,double>
  #define PI 3.14159265358979323846
  #define  mp make_pair
  #define  pb push_back
  #define EPS (1e-8)
                                          
 //   int dx[8]={0,1,0,-1,1,1,-1,-1};
   // int dy[8]={1,0,-1,0,-1,1,1,-1};
   
                                            using namespace std;
/*
                                   			class pa3{
                                            	public:
                                            	int x;
                                   				int y,z;
                                            	pa3(int x=0,int y=0,int z=0):x(x),y(y),z(z) {}
                                            	bool operator < (const pa3 &p) const{
                                            		if(x!=p.x) return x<p.x;
                                            		if(y!=p.y) return y<p.y;
                                            		 return z<p.z;
                                            		//return x != p.x ? x<p.x: y<p.y;
                                            	}
                                   				bool operator > (const pa3 &p) const{
                                            		if(x!=p.x) return x>p.x;
                                            		if(y!=p.y) return y>p.y;
                                            		 return z>p.z;
                                            		//return x != p.x ? x<p.x: y<p.y;
                                            	}
                                            	bool operator == (const pa3 &p) const{
                                            		return x==p.x && y==p.y && z==p.z;
                                            	}
                                            		bool operator != (const pa3 &p) const{
                                            			return !( x==p.x && y==p.y && z==p.z);
                                            	}
                                            
                                            };
                                            
                                            class pa4{
                                            	public:
                                            	int x;
                                            	int y,z,w;
                                            	pa4(int x=0,int y=0,int z=0,int w=0):x(x),y(y),z(z),w(w) {}
                                            	bool operator < (const pa4 &p) const{
                                            		if(x!=p.x) return x<p.x;
                                            		if(y!=p.y) return y<p.y;
                                            		if(z!=p.z)return z<p.z;
                                            		return w<p.w;
                                            		//return x != p.x ? x<p.x: y<p.y;
                                            	}
                                            	bool operator > (const pa4 &p) const{
                                            		if(x!=p.x) return x>p.x;
                                            		if(y!=p.y) return y>p.y;
                                            		if(z!=p.z)return z>p.z;
                                            		return w>p.w;
                                            		//return x != p.x ? x<p.x: y<p.y;
                                            	}
                                            	bool operator == (const pa4 &p) const{
                                            		return x==p.x && y==p.y && z==p.z &&w==p.w;
                                            	}
                                            		
                                            
                                            };
                                            class pa2{
                                            	public:
                                            	int x,y;
                                            	pa2(int x=0,int y=0):x(x),y(y) {}
                                            	pa2 operator + (pa2 p) {return pa2(x+p.x,y+p.y);}
                                            	pa2 operator - (pa2 p) {return pa2(x-p.x,y-p.y);}
                                            	bool operator < (const pa2 &p) const{
                                            		return y != p.y ? y<p.y: x<p.x;
                                            	}
                                            	bool operator > (const pa2 &p) const{
                                            		return x != p.x ? x<p.x: y<p.y;
                                            	}
                                            	bool operator == (const pa2 &p) const{
                                            		return abs(x-p.x)==0 && abs(y-p.y)==0;
                                            	}
                                            	bool operator != (const pa2 &p) const{
                                            		return !(abs(x-p.x)==0 && abs(y-p.y)==0);
                                            	}
                                            		
                                            
                                            };
                                            
 
                      
                                string itos( int i ) {
                                ostringstream s ;
                                s << i ;
                                return s.str() ;
                                }
                                 
                                int Gcd(int v,int b){
                                	if(v==0) return b;
                                	if(b==0) return v;
                                	if(v>b) return Gcd(b,v);
                                	if(v==b) return b;
                                	if(b%v==0) return v;
                                	return Gcd(v,b%v);
                                }
                 
                            
                                int mod;
int extgcd(int a, int b, int &x, int &y) {
    if (b == 0) {
        x = 1;
        y = 0;
        return a;
    }
    int d = extgcd(b, a%b, y, x);
    y -= a/b * x;
    return d;
}
pa operator+(const pa & l,const pa & r) {   
    return {l.first+r.first,l.second+r.second};                                    
}    
pa operator-(const pa & l,const pa & r) {   
    return {l.first-r.first,l.second-r.second};                                    
}  
                int beki(int wa,int rr,int warukazu){
                	if(rr==0) return 1%warukazu;
                	if(rr==1) return wa%warukazu;
                	wa%=warukazu;
                	if(rr%2==1) return ((ll)beki(wa,rr-1,warukazu)*(ll)wa)%warukazu;
                	ll zx=beki(wa,rr/2,warukazu);
                	return (zx*zx)%warukazu;
                }
 
                              
                int pr[1000100];
                int inv[1000110];
                

                
    			int comb(int nn,int rr){
    				if(rr<0 || rr>nn || nn<0) return 0;
    				int r=pr[nn]*inv[rr];
    				r%=mod;
    				r*=inv[nn-rr];
    				r%=mod;
    				return r;
    			}
               
                void gya(int ert){
                	pr[0]=1;
                	for(int i=1;i<=ert;i++){
                		pr[i]=((ll)pr[i-1]*i)%mod;
                	}
                		inv[ert]=beki(pr[ert],mod-2,mod);
                	for(int i=ert-1;i>=0;i--){
                		inv[i]=(ll)inv[i+1]*(i+1)%mod;
                	}
                }
               
              //   cin.tie(0);
    		//	ios::sync_with_stdio(false);
    			//priority_queue<pa3,vector<pa3>,greater<pa3>> pq;            
                 //sort(ve.begin(),ve.end(),greater<int>());
                 */
 //   mt19937(clock_per_sec);
  //  mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count()) ;

int co[100][100];
int dp[1<<18]={};
void solve(vector<int> ve,int b,int k){
	//int n=ve.size();
//	cout<<n<<" 
	int ans=0;
	for(int i=0;i<(1<<b);i++){
		map<int,int>ma;
		for(auto v:ve){
			ma[v&i]++;
		}
		int e=0;
		for(auto it:ma){
			for(int j=1;j<=min(it.second,k);j++)e+=co[it.second][j];
		}
		if(__builtin_popcount(i)%2)ans-=e;
		else ans+=e;
	}
	cout<<ans<<endl;
}
signed main(){
	cin.tie(0);
	ios::sync_with_stdio(false);

	int n,k,s,t;
	cin>>n>>k>>s>>t;
	if((s&t)!=s){
		cout<<0<<endl;
		return 0;
	}
	vector<int> ve;
	vector<int> W;
	for(int i=0;i<n;i++){
		int y;
		cin>>y;
		W.pb(y);
		if((s&y)==s && (t|y)==t)ve.pb(y);
	}
	if(ve.size()==0){
		cout<<0<<endl;
		return 0;
	}
	vector<int> ke;
	vector<int> ve2;
	for(int i=0;i<18;i++)if((s&(1<<i))==0)if((t&(1<<i)))ke.pb(i);
	for(auto v:ve){
		int y=0;
		for(auto w:ke){
			y*=2;
			if(v&(1<<w))y++;
		}
		ve2.pb(y);
	}
	if(ke.size()==0){
		cout<<1<<endl;
		return 0;
	}
	
	for(int i=0;i<100;i++){
		co[i][0]=1;
		co[i][i]=1;
		for(int j=1;j<i;j++)co[i][j]=co[i-1][j-1]+co[i-1][j];
	}
	//cout<<co[50][25]<<endl;
	solve(ve2,ke.size(),k);
	
	return 0;
	
	int B[30]={};
	for(int i=0;i<(1<<(ve.size()));i++){
		int g=(1<<30)-1;
		int h=0;
		for(int j=0;j<n;j++)if(i&(1<<j)){
			g&=ve[j];
			h|=ve[j];
		}
		//cout<<g<<" "<<h<<endl;
		if(g==s&&h==t)B[__builtin_popcount(i)]++;
	}
	for(int i=1;i<=ve2.size();i++)cout<<B[i]<<" ";
	cout<<endl;
		int C[30]={};
	for(int i=0;i<(1<<(n));i++){
		int g=(1<<30)-1;
		int h=0;
		for(int j=0;j<n;j++)if(i&(1<<j)){
			g&=W[j];
			h|=W[j];
		}
		//cout<<g<<" "<<h<<endl;
		if(g==s&&h==t)C[__builtin_popcount(i)]++;
	}
	for(int i=1;i<=ve2.size();i++)cout<<C[i]<<" ";
	
	//for(auto v:ve)cout<<v<<endl;
	return 0;
 }