メモ化再帰。

{{{#!highlight c++
#include <iostream>
#include <cstring>

using namespace std;

int v[8] = {200, 100, 50, 20, 10, 5, 2, 1};

int dp[8][201];
int dfs(int index, int rem) {
    if (index == 8) {
        if (rem == 0) return 1;
        return 0;
    }
    if (dp[index][rem] >= 0) return dp[index][rem];
    int sum = 0;
    for (int i = 0; i <= rem; i += v[index]) {
        sum += dfs(index+1, rem - i);
    }
    return dp[index][rem] = sum;
}

int main() {
    memset(dp, -1, sizeof(dp));
    cout << dfs(0, 200) << endl;
    return 0;
}
}}}