Saving Beans

Time Limit: 6000/3000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 8258    Accepted Submission(s): 3302

Problem Description
Although
winter is far away, squirrels have to work day and night to save beans.
They need plenty of food to get through those long cold days. After
some time the squirrel family thinks that they have to solve a problem.
They suppose that they will save beans in n different trees. However,
since the food is not sufficient nowadays, they will get no more than m
beans. They want to know that how many ways there are to save no more
than m beans (they are the same) in n trees.

Now they turn to you
for help, you should give them the answer. The result may be extremely
huge; you should output the result modulo p, because squirrels can’t
recognize large numbers.
 

 

Input
The first line contains one integer T, means the number of cases.

Then
followed T lines, each line contains three integers n, m, p, means that
squirrels will save no more than m same beans in n different trees, 1
<= n, m <= 1000000000, 1

be a prime.
 

 

Output
You should output the answer modulo p.
 

 

Sample Input
2
1 2 5
2 1 5
 

 

Sample Output
3
3
Hint

Hint For sample 1, squirrels will put no more than 2 beans in one tree. Since trees are different, we can label them as 1, 2 … and so on. The 3 ways are: put no beans, put 1 bean in tree 1 and put 2 beans in tree 1. For sample 2, the 3 ways are: put no beans, put 1 bean in tree 1 and put 1 bean in tree 2.
这题涉及乘法逆元、Lucas定理、组合数公式。
通过分析题目可以得到方案数为C(n+m-1,n-1)%p
再通过化简得：方案数为C(m+n,m)%p
 
由于考虑到T的值很大，p的值较小，我们要对1...p 的阶乘打表
另外，一定要注意0! = 1  ！！！
血的教训
 
代码：

1 #include
2 #include
3 using namespace std;
4 long long n,m,p;
5 long long sum[500050];
6
7 long long quick_mod(long long a,long long b)
8 {
9 long long ans=1;
10 while(b)
11 {
12 if(b&1)ans=ans*a%p;
13 b>>=1;
14 a=a*a%p;
15 }
16 return ans;
17 }
18
19 long long C(long long n,long long m)
20 {
21 if(n22 return (sum[n]*quick_mod((sum[m]*sum[n-m])%p,p-2)%p)%p;
23 }
24
25 long long Lucas(long long n,long long m)
26 {
27 if(m==0)return 1;
28 return C(n%p,m%p)*Lucas(n/p,m/p)%p;
29 }
30
31 int main()
32 {
33 int cas=0;scanf("%d",&cas);
34 while(cas--)
35 {
36 scanf("%I64d%I64d%I64d",&n,&m,&p);
37 sum[0]=1;for(int i=1;i<=p;i++)sum[i]=sum[i-1]*i%p;
38 printf("%I64d\n",Lucas(n+m,m));
39 }
40 return 0;
41 }
View Code

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