upc-2021个人训练赛第27场 K: ABBA (组合数学)
作者:互联网
问题 K: ABBA
时间限制: 1 Sec 内存限制: 128 MB
题目描述
Bobo has a string of length 2(n + m) which consists of characters A
and B
. The string also has a fascinating property: it can be decomposed into (n + m) subsequences of length 2, and among the (n + m) subsequences n of them are AB
while other m of them are BA
.
Given n and m, find the number of possible strings modulo (109+7).
输入
The input consists of several test cases and is terminated by end-of-file.
Each test case contains two integers n and m.
- 0≤n,m≤103
- There are at most 2019 test cases, and at most 20 of them has max{n,m}>50.
输出
For each test case, print an integer which denotes the result.
样例输入 Copy
1 2
1000 1000
0 0
样例输出 Copy
13
436240410
1
思路:
考虑用总的方案数减去不合法的方案数。
比如说第一个样例,考虑极端的情况\(AAABBB\)
这种情况是不合法的,因为\(B\)后面没有足够的\(A\)来构成\(BA\)组合。
移动一个\(A\)到任意的地方,都是不合法的,比如\(AABABB,AABBAB,AABBBA\).
也就是说,为了凑成不合法的\(BA\),最多可以将\(m-1\)的\(A\)任意排列,方案数为\(c[2n+2m][m-1]\)
另一个不合法的方案数也同上。
注意只有\(>0\)的时候才有不合法的方案数。
代码:
ll ksm(ll a,ll b,ll p){
ll res=1;
while(b){
//&运算当相应位上的数都是1时,该位取1,否则该为0。
if(b&1)
res=1ll*res*a%p;//转换为ll型
a=1ll*a*a%p;
b>>=1;//十进制下每除10整数位就退一位
}
return res;
}
const ll mod=1e9+7;
const int N=4100;
ll fact[N];//阶乘
ll infact[N];//逆元
void init(){
fact[0]=1;
infact[0]=1;
for(int i=1;i<N;i++){
fact[i]=fact[i-1]*i%mod;
infact[i]=infact[i-1]*ksm(i,mod-2,mod)%mod;
}
}
ll cul(ll a,ll b){
return fact[a]%mod*infact[b]%mod*infact[a-b]%mod;
}
void solve(){
ll n,m;
while(scanf("%lld%lld",&n,&m)!=EOF){
ll t=2*n+2*m;
ll ans=cul(t,n+m);
if(n) ans=(ans-cul(t,n-1)+mod)%mod;
if(m) ans=(ans-cul(t,m-1)+mod)%mod;
printf("%lld\n",ans);
}
}
标签:them,27,BA,ABBA,res,ll,upc,合法,test 来源: https://www.cnblogs.com/OvOq/p/15067552.html