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Problem Description
On a plain of hexagonal grid, we define a step as one move from the current grid to the lower/lower-left/lower-right grid. For example, we can move from (1,1) to (2,1), (2,2) or (3,2).
In the following graph we give a demonstrate of how this coordinate system works.
Your task is to calculate how many possible ways can you get to grid(A,B) from gird(1,1), where A and B represent the grid is on the B-th position of the A-th line.
Input
For each test case, two integers A (1<=A<=100000) and B (1<=B<=A) are given in a line, process till the end of file, the number of test cases is around 1200.
Output
For each case output one integer in a line, the number of ways to get to the destination MOD 1000000007.
Sample Input
1 1 3 2 100000 100000
Sample Output
1 3 1
Hint
Source
“浪潮杯”山东省第八届ACM大学生程序设计竞赛(感谢青岛科技大学)
思路:若只能向↙或↘走,从(1,1)->(x,y),总共要走tot=(x-1)步,其中向↘要走r=(y-1)步,向↙要走l=(x-y)步,那么总的走法就是C(tot,r)(或C(tot,l))。 若还能向↓走,因为一步↓相当于一步↙加上一步↘,所以若向↓走i步, 总共就只要走tot'=(tot-i)步,向↘走r'=(r-i)步,向↙走l'=(l-i)步,(其中0<=i<=min(l,r)), 那么总的走法数就是C(tot',i)+C(tot'-i,r)(表达形式不止一种)。
code:
#include <bits/stdc++.h> using namespace std; const int mod = 1e9+7; typedef long long ll; ll fac[110000]; void init(){ fac[0] = fac[1] = 1; for(int i = 2; i <= 100000; i++){ fac[i] = (fac[i-1] * i) % mod; } } ll q_pow(ll a,ll b){ ll ans = 1; while(b){ if(b & 1) ans = ans * a % mod; b >>= 1; a = a * a % mod; } return ans % mod; } ll C(ll n,ll m){ if(m > n) return 0; return fac[n] * q_pow(fac[m] * fac[n-m], mod - 2) % mod; } ll Lucas(ll n,ll m){ if(m == 0) return 1; else return (C(n%mod,m%mod) * Lucas(n/mod,m/mod)) % mod; } int main(){ init(); ll x,y; while(~scanf("%lld%lld",&x,&y)){ ll all = x - 1; ll r1 = y - 1; ll l1 = x - y; ll down = min(l1,r1); ll ans = 0; for(ll i = 0; i <= down; i++){ ll tot = all - i; ll l = l1 - i; ll r = r1 - i; ans = (ans + Lucas(tot,i) % mod * Lucas(tot-i,l) % mod) % mod; } printf("%lld\n",ans); } return 0; }