There are $500^7$ different combinations to check, which is far too many.

However, just like with the bronze version of this problem, where we were only concerned about the parity of the answer, we are only concerned with the result of the product mod 7, so we only care about the values of the variables mod 7. This gives us $7^7$ different combinations to check, which will run in time.

Here is Mark Gordon's code.

#include <iostream>
#include <cstdio>

using namespace std;

long long num[256][7];

int main() {
  freopen("bgm.in", "r", stdin);
  freopen("bgm.out", "w", stdout);

  int N;
  cin >> N;

  for (int i = 0; i < N; i++) {
    char letter;
    int val;
    cin >> letter >> val;
    num[letter][(val % 7 + 7) % 7]++;
  }

  long long result = 0;

  /* Try every possible residue mod 7 for the variables. */
  for(int B = 0; B < 7; B++)
  for(int E = 0; E < 7; E++)
  for(int S = 0; S < 7; S++)
  for(int I = 0; I < 7; I++)
  for(int G = 0; G < 7; G++)
  for(int O = 0; O < 7; O++)
  for(int M = 0; M < 7; M++) {
    if (((B + E + S + S + I + E) * (G + O + E + S) * (M + O + O)) % 7 == 0) {
      result += num['B'][B] * num['E'][E] * num['S'][S] * num['I'][I] *
                num['G'][G] * num['O'][O] * num['M'][M];
    }
  }
  cout << result << endl;

  return 0;
}