题目链接：zoj 3822 Domination

题目大意：给定一个N?M的棋盘，每次任选一个位置放置一枚棋子，直到每行每列上都至少有一枚棋子，问放置棋子个数的期望。

解题思路：大白书上概率那一张有一道类似的题目，但是因为时间比较久了，还是稍微想了一下。
dp[i][j][k]表示i行j列上均有至少一枚棋子，并且消耗k步的概率（k≤i?j）,因为放置在i+1~n上等价与放在i+1行上，同理列也是如此。所以有转移方程：

• dp[i][j][k+1]+=dp[i][j][k]?(n?k)(S?k)
• dp[i+1][j][k+1]+=dp[i][j][k]?(N?i)?j(S?k)
• dp[i][j+1][k+1]+=dp[i][j][k]?(M?j)?i(S?k)
• dp[i+1][j+1][k+1]+=dp[i][j][k]?(N?i)?(M?j)(S?k)

#include <cstdio>
#include <cstring>
#include <algorithm>

using namespace std;

const int maxn = 55;
const int maxm = 2505;

int N, M;
double dp[maxn][maxn][maxm];

double solve () {
    int S = N * M;
    memset(dp, 0, sizeof(dp));
    dp[1][1][1] = 1;
    for (int i = 1; i <= N; i++) {
        for (int j = 1; j <= M; j++) {
            int n = i * j;
            for (int k = max(i, j); k <= n; k++) {
                dp[i][j][k+1] += dp[i][j][k] * (n - k) / (S - k);
                dp[i+1][j][k+1] += dp[i][j][k] * (N - i) * j / (S - k);
                dp[i][j+1][k+1] += dp[i][j][k] * (M - j) * i / (S - k);
                dp[i+1][j+1][k+1] += dp[i][j][k] * (N - i) * (M - j) / (S - k);
            }
        }
    }

    /*
    for (int i = 1; i <= N; i++) {
        for (int j = 1; j <= M; j++) {
            printf("%d %d:", i, j);
            for (int k = max(i, j); k <= i * j; k++)
                printf("%.3lf ", dp[i][j][k]);
            printf("\n");
        }
    }
    */

    double ans = 0;
    for (int i = max(N, M); i <= S; i++)
        ans += (dp[N][M][i] - dp[N][M][i-1]) * i;
    return ans;
}

int main () {
    int cas;
    scanf("%d", &cas);
    while (scanf("%d%d", &N, &M) == 2) {
        printf("%.8lf\n", solve());
    }
    return 0;
}

zoj 3822 Domination(dp)