【题目】

Killer Names

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others)
Total Submission(s): 1898 Accepted Submission(s): 924

Problem Description

> Galen Marek, codenamed Starkiller, was a male Human apprentice of the Sith Lord Darth Vader. A powerful Force-user who lived during the era of the Galactic Empire, Marek originated from the Wookiee home planet of Kashyyyk as the sole offspring of two Jedi Knights—Mallie and Kento Marek—who deserted the Jedi Order during the Clone Wars. Following the death of his mother, the young Marek's father was killed in battle by Darth Vader. Though only a child, Marek possessed an exceptionally strong connection to the Force that the Dark Lord of the Sith sought to exploit.
>
> When Marek died in 2 BBY, shortly after the formation of the Alliance, Vader endeavored to recreate his disciple by utilizing the cloning technologies of the planet Kamino. The accelerated cloning process—an enhanced version of the Kaminoan method which allowed for a rapid growth rate within its subjects—was initially imperfect and many clones were too unstable to take Marek's place as the Dark Lord's new apprentice. After months of failure, one particular clone impressed Vader enough for him to hope that this version might become the first success. But as with the others, he inherited Marek's power and skills at the cost of receiving his emotions as well, a side effect of memory flashes used in the training process.
>
> — Wookieepedia

Darth Vader is finally able to stably clone the most powerful soilder in the galaxy: the Starkiller. It is the time of the final strike to destroy the Jedi remnants hidden in every corner of the galaxy.

However, as the clone army is growing, giving them names becomes a trouble. A clone of Starkiller will be given a two-word name, a first name and a last name. Both the first name and the last name have exactly n characters, while each character is chosen from an alphabet of size m . It appears that there are m2n possible names to be used.

Though the clone process succeeded, the moods of Starkiller clones seem not quite stable. Once an unsatisfactory name is given, a clone will become unstable and will try to fight against his own master. A name is safe if and only if no character appears in both the first name and the last name.

Since no two clones can share a name, Darth Vader would like to know the maximum number of clones he is able to create.

Input

The First line of the input contains an integer T (T≤10 ), denoting the number of test cases.

Each test case contains two integers n and m (1≤n,m≤2000 ).

Output

For each test case, output one line containing the maximum number of clones Vader can create.

Output the answer mod 109+7

Sample Input

3 2

2 3

Sample Output

【题解】

题意：给定m种不同字母，要求从m种字母中选出并组成合法名字，当任意字母不同时出现在姓和名中时称这个名字是合法的，姓和名的长度均为n，问最多可以组成多少个合法的名字。

思路：相当于涂色，给定m种颜色，为两个长度为n的单元格涂色，要求两个区域没有共同的颜色。这个题的方法就是先考虑姓，得到长度为n的分别用了j=1到m种颜色的方案数，那么对于名，就是(m-j)^n，即剩下的颜色填到n个格子的方案数。

dp[i][j]表示长度为i，用到j种颜色的情况数，显然dp[1][1]=m，状态转移方程为：dp[i][j]=dp[i-1][j]*j+dp[i-1][j-1]*(m-(j-1))。

长度为i用到j种颜色的数目=长度为i-1用到j种颜色的数目*j种颜色（第i个格是j种颜色中的一种）+长度为i-1用到j-1种颜色*（m-(j-1))种颜色（第i个格用的是剩下的m-(j-1)种颜色）。

【代码】

#include<stdio.h>
typedef long long ll;
const ll mod=1e9+7;
ll pow(ll x,ll n)
{
    ll res=1;
    while(n>0)
    {
        if(n&1) res=res*x%mod;
        x=x*x%mod;
        n>>=1;
    }
    return res;
}
ll dp[2005][2005]; //表示当长度为i时使用j种颜色的情况数目
main()
{
    int T,n,m; scanf("%d",&T);
    while(T--)
    {
        scanf("%d%d",&n,&m);
        dp[1][1]=m;
        for(int i=2;i<=n;i++)
            for(int j=1;j<=i&&j<=m;j++)
                dp[i][j]=dp[i-1][j]*j%mod+dp[i-1][j-1]*(m-(j-1))%mod;
        ll ans=0;
        for(int j=1;j<=m;j++)
            ans=(ans+dp[n][j]*pow(m-j,n)%mod)%mod;
        printf("%lld\n",ans);
    }
}

hdu 6143 Killer Names（dp+思维）

Killer Names

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