1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
#include<bits/stdc++.h>
using namespace std;
using ll=long long;
struct Mod
{
	ll m,p;
	void init(int pp){m=((__int128)1<<64)/pp;p=pp;}
	ll operator ()(ll x)
	{
		ll ret=x-((__int128(x)*m)>>64)*p;
		if(ret>=p) ret-=p;
		return ret;
	}
}mod;
ll ksm(ll a,ll b)
{
	ll ans=1;
	while(b)
	{
		if(b&1) ans=mod(ans*a);
		b>>=1;
		a=mod(a*a);
	}
	return ans;
}
ll p;
ll fact[3001000],rfact[3001000],inv[3001000];
void init(int n)
{
	fact[0]=rfact[0]=1;
	inv[1]=1;
	for(int i=2;i<=n;i++)
		inv[i]=mod((p-p/i)*inv[p%i]);
	for(int i=1;i<=n;i++)
	{
		fact[i]=mod(fact[i-1]*i);
		rfact[i]=mod(rfact[i-1]*inv[i]);
	}
}
ll standard_young_tableaux(vector<int> mu)
{
	int n=mu.size();
	int m=mu[0];
	vector<vector<int>> grid(n,vector<int>(m));
	vector<vector<int>> arm(n+1,vector<int>(m+1)),leg(n+1,vector<int>(m+1));
	for(int i=0;i<n;i++)
		for(int j=0;j<mu[i];j++)
			grid[i][j]=1;
	for(int i=n-1;i>=0;i--)
		for(int j=m-1;j>=0;j--)
		{
			arm[i][j]=arm[i+1][j]+1;
			leg[i][j]=leg[i][j+1]+1;
			if(!grid[i][j])
				arm[i][j]=leg[i][j]=0;
		}
	int tot=accumulate(mu.begin(),mu.end(),0);
	ll ret=fact[tot];
	for(int i=0;i<n;i++)
		for(int j=0;j<m;j++)
			if(grid[i][j])
				ret=mod(ret*inv[arm[i][j]+leg[i][j]-1]);
	return ret;
}
ll E(int r,int p,int s)
{
	if(r&1)
	{
		ll ret=E(r-1,p,s);
		ret=mod(ret*fact[r/2+s]);
		ret=mod(ret*rfact[p-r/2+s]);
		return ret;
	}
	ll S1=1,S2=1;
	for(int l=r+1;l<p+p-r+2;l++)
		S1=mod(S1*ksm(inv[l+s+s],r/2));
	for(int l=2;l<=r;l++)
		S2=mod(S2*ksm(mod(inv[l+s+s]*inv[p+p-l+s+s+2]),l/2));
	return mod(S1*S2);
}
ll shifted_triangle_tableaux(int n)
{
	ll ret=fact[n*(n+1)/2];
	for(int i=0;i<n;i++)
		ret=mod(mod(ret*fact[i])*rfact[i+i+1]);
	return ret;
}
ll truncated_triangle_tableaux(int n,int m,int k)
{
	if(n<m) swap(n,m);
	if(n==m&&n==k+1) return shifted_triangle_tableaux(n);
	int N=n*m-k*(k+1)/2;
	ll ret=1;
	ret=mod(ret*fact[N]);
	ret=mod(ret*rfact[m*(n-k-1)]);
	ret=mod(ret*rfact[N-m*(n-k-1)]);
	ret=mod(ret*standard_young_tableaux(vector<int>(m,n-k-1)));
	ret=mod(ret*truncated_triangle_tableaux(m,k+1,k));
	ret=mod(ret*E(k+1,m,n-k-1));
	ret=mod(ret*ksm(E(k+1,m,0),p-2));
	return ret;
}
int main()
{
	ios::sync_with_stdio(false);
	cin.tie(0);
	cout.tie(0);
	int a,b,c;
	cin>>a>>b>>c;
	cin>>p;
	mod.init(p);
	init(3000000);
	if(c==a+b-1)
	{
		ll tot=fact[a*b];
		for(int i=1;i<=a;i++)
			for(int j=1;j<=b;j++)
				tot=mod(tot*inv[i+j-1]);
		cout<<a*b<<" "<<mod(tot*tot)<<'\n';
		return 0;
	}
	int ans1=0;
	for(int i=1;i<=a;i++)
		for(int j=1;j<=b;j++)
			if(i+j<=c+1)
				ans1++;
	vector<int> mu(a);
	for(int i=0;i<a;i++)
		mu[i]=min(b,c-i);
	ll ans2=standard_young_tableaux(mu);
	ans2=mod(ans2*truncated_triangle_tableaux(a,b,a+b-c-1));
	cout<<ans1<<" "<<ans2<<endl;
	return 0;
}