ZS the Coder has recently found an interesting concept called the Birthday Paradox. It states that given a random set of 23 people, there is around 50% chance that some two of them share the same birthday. ZS the Coder finds this very interesting, and decides to test this with the inhabitants of Udayland.

In Udayland, there are 2n days in a year. ZS the Coder wants to interview k people from Udayland, each of them has birthday in one of 2n days (each day with equal probability). He is interested in the probability of at least two of them have the birthday at the same day.

ZS the Coder knows that the answer can be written as an irreducible fraction . He wants to find the values of A and B (he does not like to deal with floating point numbers). Can you help him?

Input

The first and only line of the input contains two integers n and k (1 ≤ n ≤ 1018, 2 ≤ k ≤ 1018), meaning that there are 2n days in a year and that ZS the Coder wants to interview exactly k people.

Output

If the probability of at least two k people having the same birthday in 2n days long year equals (A ≥ 0, B ≥ 1, ), print the A and B in a single line.

Since these numbers may be too large, print them modulo 106 + 3. Note that A and B must be coprime before their remainders modulo 106 + 3 are taken.

Examples
Input
3 2
Output
1 8
Input
1 3
Output
1 1
Input
4 3
Output
23 128
Note

In the first sample case, there are 23 = 8 days in Udayland. The probability that 2 people have the same birthday among 2 people is clearly , so A = 1, B = 8.

In the second sample case, there are only 21 = 2 days in Udayland, but there are 3 people, so it is guaranteed that two of them have the same birthday. Thus, the probability is 1 and A = B = 1.

想要知道答案是啥还是很容易的,用高中知识即可知道1 - {  A(2^n,k) / 2^(nk)  } 即是所求,A(a,b)是排列数。

问题是要先化简再上下同时取模

可以先证明如果有大数A、B,假设(A%mod)/(B%mod)==p/q,那么1-A/B=(B-A)/B=(q-p)/q。(在模mod意义下)

所以只要知道A(2^n,k)/2^(nk),就能知道答案了

先化简。

分式下面只有2,所以gcd=2^t,t不知道,但是显然t是由上面A(2^n,k)决定。

考虑(2^n)(2^n-1)...(2^n-k+1)有多少个因子2。如果把2^n单独考虑,剩下(2^n-1)...(2^n-k+1)的2的因子数跟1~k-1的2的因子数一样多。因为任取个(2^n-s),它能和s对应

所以因子数就是n+(k-1)/2+(k-1)/4+...+(k-1)/2^p,除到(k-1)/2^p=0为止

得到了gcd=2^t的t之后,只要上下同时乘(2关于mod的逆元)乘t次,就完成了化简。

然后注意到当k>=mod时,(2^n)(2^n-1)(2^n-mod+1)%mod==0,由鸽巢原理这是显然的。

所以当k<mod,A(2^n,k)%mod暴力算,k>=mod,A(2^n,k)%mod==0。

而2^(nk)%mod是容易算的。

 #include<cstdio>
#include<iostream>
#include<cstring>
#include<cstdlib>
#include<algorithm>
#include<cmath>
#include<queue>
#include<deque>
#include<set>
#include<map>
#include<ctime>
#define LL long long
#define inf 0x7ffffff
#define pa pair<int,int>
#define mkp(a,b) make_pair(a,b)
#define pi 3.1415926535897932384626433832795028841971
#define mod 1000003
using namespace std;
inline LL read()
{
LL x=,f=;char ch=getchar();
while(ch<''||ch>''){if(ch=='-')f=-;ch=getchar();}
while(ch>=''&&ch<=''){x=x*+ch-'';ch=getchar();}
return x*f;
}
LL n,k;
LL rev_2;
LL bit[];
inline LL quickpow(LL a,LL b,LL MOD)
{
LL s=;
while (b)
{
if (b&)s=(s*a)%MOD;
a=(a*a)%MOD;
b>>=;
}
return s;
}
int main()
{
rev_2=quickpow(,mod-,mod);bit[]=;for (int i=;i<;i++)bit[i]=bit[i-]<<;
n=read();k=read();
if (n<=&&k>bit[n]){puts("1 1");return ;}
if (k==){puts("0 1");return ;}
LL mx=quickpow(,n,mod),now=mx;
LL ans1=,ans2=;
LL te=k-,sum=n;
while (te)
{
sum+=te/;
te>>=;
}
LL sv=k;
for (LL i=;i<=min(sv,(LL)mod);i++)
{
k--;
ans2=(ans2*mx)%mod;
ans1=(ans1*now)%mod;
now--;if (!now)now+=mod;
}
while (k%(mod-)!=)ans2=(ans2*mx)%mod,k--;
ans1=(ans1*quickpow(rev_2,sum,mod))%mod;
ans2=(ans2*quickpow(rev_2,sum,mod))%mod;
printf("%lld %lld\n",(ans2-ans1+mod)%mod,ans2);
//ans=1-{ (2^n*(2^n-1)*(2^n-2)*...*(2^n-k+1))/(2^n)^k }
}

cf 711E

cf711E ZS and The Birthday Paradox的更多相关文章

  1. CF369E. ZS and The Birthday Paradox

    /* cf369E. ZS and The Birthday Paradox http://codeforces.com/contest/711/problem/E 抽屉原理+快速幂+逆元+勒让德定理 ...

  2. codeforces 711E E. ZS and The Birthday Paradox(数学+概率)

    题目链接: E. ZS and The Birthday Paradox. time limit per test 2 seconds memory limit per test 256 megaby ...

  3. ZS and The Birthday Paradox

    ZS and The Birthday Paradox 题目链接:http://codeforces.com/contest/711/problem/E 数学题(Legendre's formula) ...

  4. Codeforces 711E ZS and The Birthday Paradox 数学

    ZS and The Birthday Paradox 感觉里面有好多技巧.. #include<bits/stdc++.h> #define LL long long #define f ...

  5. Codeforces Round #369 (Div. 2) E. ZS and The Birthday Paradox 数学

    E. ZS and The Birthday Paradox 题目连接: http://www.codeforces.com/contest/711/problem/E Description ZS ...

  6. 【Codeforces711E】ZS and The Birthday Paradox [数论]

    ZS and The Birthday Paradox Time Limit: 20 Sec  Memory Limit: 512 MB Description Input Output Sample ...

  7. Codeforces 711E ZS and The Birthday Paradox

    传送门 time limit per test 2 seconds memory limit per test 256 megabytes input standard input output st ...

  8. 【28.57%】【codeforces 711E】ZS and The Birthday Paradox

    time limit per test2 seconds memory limit per test256 megabytes inputstandard input outputstandard o ...

  9. codeforces 711E. ZS and The Birthday Paradox 概率

    已知一年365天找23个人有2个人在同一天生日的概率 > 50% 给出n,k ,表示现在一年有2^n天,找k个人,有2个人在同一天生日的概率,求出来的概率是a/b形式,化到最简形式,由于a,b可 ...

随机推荐

  1. selenium-Python之鼠标事件

    通过click()来模拟鼠标的单击操作,鼠标还具有鼠标右击,双击,悬停甚至鼠标拖动等功能.在webdriver中,将这些鼠标操作方法封装在ActionChains类提供. ActionChains类提 ...

  2. ubuntu 14.04 安装redis

    root@hett-PowerEdge-T30:~# sudo apt-get install redis-server Reading package lists... DoneBuilding d ...

  3. navicate与mysql连接的中文乱码问题

    1. 在navicate中查看 show variables like'char%'; show variables like 'collation_%'; 2.在mysql中查看 通过对比可以发现两 ...

  4. 用Hexo免费搭建你自己的博客

    Hexo基于node.js,可用于生成静态博客,结合github和Mac,可以专注创作了. 深入学习见文末引用. hexo安装 brew install node npm install hexo-c ...

  5. UVA10917 A walk trough the Forest (最短路,dp)

    求出家到其他点的最短路径,题目的条件变成了u->v不是回头路等价于d[u]>d[v]. 然后根据这个条件建DAG图,跑dp统计方案数,dp[u] = sum(dp[v]). #includ ...

  6. WPF知识点全攻略08- 依赖属性

    依赖属性是WPF不得不提,不得不会系列又一 先来看一下,自定义依赖属性的写法 public static readonly DependencyProperty IconProperty = Depe ...

  7. Spring启动流程—源码解读

    https://blog.csdn.net/yangliuhbhd/article/details/80790761 Spring的AbstractApplicationContext的refresh ...

  8. Bootstrap历练实例:表单控件状态(禁用的字段集fieldset)

    禁用的字段集 fieldset 对 <fieldset> 添加 disabled 属性来禁用 <fieldset> 内的所有控件. <!DOCTYPE html>& ...

  9. WinForm各种关闭

    Appication.Exit(); Environment.Exit(); System.Threading.Thread.CurrentThread.Abort(); Process.GetCur ...

  10. java在线聊天项目0.3版本 制作客户端窗体,实现发送按钮和回车发送信息功能,使用ActionListener监听事件中actionPerformed方法(用内部类和匿名内部类两种方法)

    方法一,使用匿名内部类的监听方法,因方法一致代码稍冗余 package com.swift; import java.awt.BorderLayout; import java.awt.Color; ...