题意：给一个矩阵a，a[i][j] = C[i][j](i>=j) or 0(i < j)，求(x1,y1)，(x2,y2)这个子矩阵里面的所有数的和。
思路：首先问题可以转化为求(0,0)，(n,m)这个子矩阵的所有数之和。画个图容易得到一个做法，对于n<=m，答案就是2^0+2^1+...+2^m=2^(m+1)-1，对于n>m，答案由两部分构成，一部分是2^(m+1)-1，另一部分是sigma i:m+1->n f[i][m]，f[i][m]表示第i行前m列的数之和，f数组存在如下关系，f[i][m]=f[i-1][m]*2-C[i-1][m]，f[m][m]=2^m。还有另一种思路：第i列的所有数之和为C(i,i)+C(i+1,i)+...+C(n,i)=C(n+1,i+1)，于是答案就是sigma i:0->min(n,m) C(n+1,i+1)。
Lucas定理：由于题目给定的模是可变的质数，且质数可能很小，那么就不能直接用阶乘和阶乘的逆相乘了，需要用到Lucas定理，公式：C(n,m)%P=C(n/P,m/P)*C(n%P,m%P)%P，c(n,m)=0(n<m)。当然最终还是要预处理阶乘和阶乘的逆来得到答案。复杂度O(nlogP+nlogn)
下面是第一种思路的代码：
#pragma comment(linker, "/STACK:10240000,10240000") #include <iostream>
#include <cstdio>
#include <algorithm>
#include <cstdlib>
#include <cstring>
#include <map>
#include <queue>
#include <deque>
#include <cmath>
#include <vector>
#include <ctime>
#include <cctype>
#include <set>
#include <bitset>
#include <functional>
#include <numeric>
#include <stdexcept>
#include <utility> using namespace std; #define mem0(a) memset(a, 0, sizeof(a))
#define mem_1(a) memset(a, -1, sizeof(a))
#define lson l, m, rt << 1
#define rson m + 1, r, rt << 1 | 1
#define define_m int m = (l + r) >> 1
#define rep_up0(a, b) for (int a = 0; a < (b); a++)
#define rep_up1(a, b) for (int a = 1; a <= (b); a++)
#define rep_down0(a, b) for (int a = b - 1; a >= 0; a--)
#define rep_down1(a, b) for (int a = b; a > 0; a--)
#define all(a) (a).begin(), (a).end()
#define lowbit(x) ((x) & (-(x)))
#define constructInt4(name, a, b, c, d) name(int a = 0, int b = 0, int c = 0, int d = 0): a(a), b(b), c(c), d(d) {}
#define constructInt3(name, a, b, c) name(int a = 0, int b = 0, int c = 0): a(a), b(b), c(c) {}
#define constructInt2(name, a, b) name(int a = 0, int b = 0): a(a), b(b) {}
#define pchr(a) putchar(a)
#define pstr(a) printf("%s", a)
#define sstr(a) scanf("%s", a)
#define sint(a) scanf("%d", &a)
#define sint2(a, b) scanf("%d%d", &a, &b)
#define sint3(a, b, c) scanf("%d%d%d", &a, &b, &c)
#define pint(a) printf("%d\n", a)
#define test_print1(a) cout << "var1 = " << a << endl
#define test_print2(a, b) cout << "var1 = " << a << ", var2 = " << b << endl
#define test_print3(a, b, c) cout << "var1 = " << a << ", var2 = " << b << ", var3 = " << c << endl
#define mp(a, b) make_pair(a, b)
#define pb(a) push_back(a) typedef unsigned int uint;
typedef long long LL;
typedef pair<int, int> pii;
typedef vector<int> vi; const int dx[] = {, , -, , , , -, -};
const int dy[] = {-, , , , , -, , - };
const int maxn = 1e8 + ;
const int md = 1e9 + ;
const int inf = 1e9 + ;
const LL inf_L = 1e18 + ;
const double pi = acos(-1.0);
const double eps = 1e-; template<class T>T gcd(T a, T b){return b==?a:gcd(b,a%b);}
template<class T>bool max_update(T &a,const T &b){if(b>a){a = b; return true;}return false;}
template<class T>bool min_update(T &a,const T &b){if(b<a){a = b; return true;}return false;}
template<class T>T condition(bool f, T a, T b){return f?a:b;}
template<class T>void copy_arr(T a[], T b[], int n){rep_up0(i,n)a[i]=b[i];}
int make_id(int x, int y, int n) { return x * n + y; } struct ModInt {
static int MD;
int x;
ModInt(int xx = ) { if (xx >= ) x = xx % MD; else x = MD - (-xx) % MD; }
int get() { return x; } ModInt operator + (const ModInt &that) const { int x0 = x + that.x; return ModInt(x0 < MD? x0 : x0 - MD); }
ModInt operator - (const ModInt &that) const { int x0 = x - that.x; return ModInt(x0 < MD? x0 + MD : x0); }
ModInt operator * (const ModInt &that) const { return ModInt((long long)x * that.x % MD); }
ModInt operator / (const ModInt &that) const { return *this * that.inverse(); } ModInt operator += (const ModInt &that) { x += that.x; if (x >= MD) x -= MD; }
ModInt operator -= (const ModInt &that) { x -= that.x; if (x < ) x += MD; }
ModInt operator *= (const ModInt &that) { x = (long long)x * that.x % MD; }
ModInt operator /= (const ModInt &that) { *this = *this / that; } ModInt inverse() const {
int a = x, b = MD, u = , v = ;
while(b) {
int t = a / b;
a -= t * b; std::swap(a, b);
u -= t * v; std::swap(u, v);
}
if(u < ) u += MD;
return u;
} };
int ModInt::MD;
int p;
#define mint ModInt mint C(mint fact[], mint fact_inv[], int n, int m) {
if (n < m) return ;
if (n < p) return fact[n] * fact_inv[m] * fact_inv[n - m];
return C(fact, fact_inv, n / p, m / p) * C(fact, fact_inv, n % p, m % p);
} mint get(mint a[], mint fact[], mint fact_inv[], int n, int m) {
if (n < || m < ) return ;
if (n <= m) return a[n + ] - ;
mint ans = a[m + ] - ;
mint last = a[m];
rep_up1(i, n - m) {
int u = m + i;
last = last * - C(fact, fact_inv, u - , m);
ans += last;
}
return ans;
} int main() {
//freopen("in.txt", "r", stdin);
int a, b, c, d;
while (cin >> a >> b >> c >> d >> p) {
mint::MD = p;
mint x = ;
mint mi2[], fact[], fact_inv[];
mi2[] = fact[] = fact_inv[] = ;
rep_up1(i, ) {
mi2[i] = mi2[i - ] * ;
fact[i] = fact[i - ] * i;
fact_inv[i] = fact_inv[i - ] / i;
}
cout << (get(mi2, fact, fact_inv, c, d) - get(mi2, fact, fact_inv, a - , d) -
get(mi2, fact, fact_inv, c, b - ) + get(mi2, fact, fact_inv, a - , b - )).get() << endl;
}
return ;
}