UVALive

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f(k)=k∗(k−1)n−1=选择不超过k种颜色,相邻颜色不同,染n个格子的方案数
(第一个格子有k种选择,之后n−1个格子有k−1种颜色可以选)
恰好使用了k种颜色的方案数=f(k)−(不用颜色1+不用颜色2+...)+(不用颜色1和2+不用颜色1和3+...)−...
ans=f(k)−Ck−1kf(k−1)+Ck−2kf(k−2)−....
ans=f(k)+∑k−1i=1(−1)iCk−ikf(k−i)

#include<stdio.h>#include<bits/stdc++.h>#define ll long long#define pii pair<int,int>#define pll pair<ll,ll>#define MEM(a,x) memset(a,x,sizeof(a))#define lowbit(x) ((x)&-(x))using namespace std;const int inf=1e9+7;const int N =  1e6+5;const int M = 5e5;ll qPow(ll a,ll n){    ll ans=1;    ll t=a%inf;    while(n){        if(n&1){            ans=(ans*t)%inf;        }        t=(t*t)%inf;        n>>=1;    }    return ans;}ll ny(ll a){//逆元    return qPow(a,inf-2);}ll fact[N];void initFact(){    fact[0]=1;    for(int i=1;i<N;++i){        fact[i]=((ll)i*fact[i-1])%inf;    }}ll C(int n,int k){    if(k>n){        return 0;    }    ll fz=fact[n];    ll fm=(fact[k]*fact[n-k])%inf;    fm=ny(fm);    return (fz*fm)%inf;}ll Cb(int n,int k){    if(k>n){        return 0;    }    ll ans=1;    for(int i=0;i<k;++i){        ans=(ans*(n-i))%inf;    }    ll fm=fact[k];    fm=ny(fm);    return (ans*fm)%inf;}ll f(ll k,int n){    ll t=(k*qPow(k-1,n-1))%inf;    return t;}ll slove(int n,int m,int k){    ll ans=f(k,n);    int flag=-1;    for(int i=1;i<k;++i){        ll t=(C(k,k-i)*f(k-i,n))%inf;        ans=(ans+flag*t)%inf;        flag*=-1;    }    return ((ans*Cb(m,k))%inf+inf)%inf;}int main(){    //freopen("/home/lu/code/r.txt","r",stdin);    //freopen("/home/lu/code/w.txt","w",stdout);    int T;    initFact();    scanf("%d",&T);    for(int tt=1;tt<=T;++tt){        int n,m,k;        scanf("%d%d%d",&n,&m,&k);        ll ans=slove(n,m,k);        printf("Case #%d: %lld\n",tt,ans);    }    return 0;}