(Analysis by Nick Wu)

We list out two dynamic programming approaches here, one that is $O(NMK)$ and one that is $O((N+M)K^2)$.

We first note that there is an $O(N^2 M^2 K)$ dynamic programming algorithm. Let $f(N, M, K)$ be the number of ways to match $K$ pairs first $N$ of Farmer John's cows and first $M$ of Farmer Paul's cows such that Farmer John wins. We can reduce $K$ by one if we fix the highest scoring cow that both Farmer John and Farmer Paul use. With $O(NM)$ transitions and $O(NMK)$ states, we get an $O(N^2 M^2 K)$ algorithm.

Instead of iterating over all $O(NM)$ transitions, if we maintain a prefix sum, we can compute $f(N, M, K)$ in $O(1)$ time. My Java code does this as below.

import java.io.*;
import java.util.*;
public class team {
public static void main(String[] args) throws IOException {
PrintWriter pw = new PrintWriter(new BufferedWriter(new FileWriter("team.out")));
final int MOD = 1000000009;
int n = Integer.parseInt(st.nextToken());
int m = Integer.parseInt(st.nextToken());
int k = Integer.parseInt(st.nextToken());
int[] a = new int[n];
int[] b = new int[m];
for(int i = 0; i < n; i++) {
a[i] = Integer.parseInt(st.nextToken());
}
for(int i = 0; i < m; i++) {
b[i] = Integer.parseInt(st.nextToken());
}
long[][] dp = new long[n+1][m+1];
for(int i = 0; i < dp.length; i++) {
Arrays.fill(dp[i], 1);
}
Arrays.sort(a);
Arrays.sort(b);
while(k-- > 0) {
long[][] next = new long[n+1][m+1];
for(int i = 0; i < a.length; i++) {
for(int j = 0; j < b.length; j++) {
if(a[i] <= b[j]) continue;
next[i+1][j+1] += dp[i][j];
}
}
dp = next;
for(int i = 0; i < dp.length; i++) {
for(int j = 0; j < dp[i].length; j++) {
if(i > 0) {
dp[i][j] += dp[i-1][j];
}
if(j > 0) {
dp[i][j] += dp[i][j-1];
if(i > 0) {
dp[i][j] -= dp[i-1][j-1];
}
}
dp[i][j] %= MOD;
dp[i][j] += MOD;
dp[i][j] %= MOD;
}
}
}
pw.println(dp[n][m] % MOD);
pw.close();
}
}


Another way to reduce the state space is to leverage the fact that $K$ is small.

Instead of explicitly pairing the highest-scoring cows that Farmer John and Farmer Paul use, we will instead iterate over the cows from lowest score to highest score. We keep track of how many cows from Farmer John we have allocated and how many cows from Farmer John we have allocated. We require that we have never allocated more cows from Farmer John than from Farmer Paul (otherwise Farmer John would have selected a cow that will not win).

This DP has $O((N+M)K^2)$ states and $O(1)$ transitions. Here is Travis Hance's C++ solution.

#include <cstdio>
#include <vector>
#include <algorithm>
using namespace std;

struct Node {
int value;
bool isFj;

bool operator<(Node const& other) const {
if (value == other.value) {
return isFj && !other.isFj;
}
return value < other.value;
}
};

#define NMAX 1000
#define KMAX 10

#define MOD 1000000009

unsigned int dp[2 * NMAX + 1][KMAX + 1][KMAX + 1];

int main() {
int n, m, k;
scanf("%d", &n);
scanf("%d", &m);
scanf("%d", &k);

vector<Node> nodes;
for (int i = 0; i < n; i++) {
Node n;
scanf("%d", &n.value);
n.isFj = true;
nodes.push_back(n);
}
for (int i = 0; i < m; i++) {
Node n;
scanf("%d", &n.value);
n.isFj = false;
nodes.push_back(n);
}

sort(nodes.begin(), nodes.end());

for (int i = 0; i <= k; i++) {
for (int j = 0; j <= k; j++) {
dp[nodes.size()][i][j] = (i == 0 && j == 0 ? 1 : 0);
}
}

for (int pos = nodes.size() - 1; pos >= 0; pos--) {
for (int i = 0; i <= k; i++) {
for (int j = 0; j <= k; j++) {
if (j > i) {
dp[pos][i][j] = 0;
} else {
if (nodes[pos].isFj) {
dp[pos][i][j] = (dp[pos+1][i][j] + (i > 0 ? dp[pos+1][i-1][j] : 0)) % MOD;
} else {
dp[pos][i][j] = (dp[pos+1][i][j] + (j > 0 ? dp[pos+1][i][j-1] : 0)) % MOD;
}
}
}
}
}

printf("%d\n", dp[k][k]);
}