Robot nxn board. Can only go right or go down. How many paths?

robot.cpp

int main() {
   rbtree<int> tree;
   int n = 10;
   vector<int> cur(n), prev(n,1);
   vector<vector<int>> board(n, vector<int>(n));

   ///
   //iota(prev.begin(), prev.end(), 0);
   cout <<endl;

   for( int i = 0; i < n; i++){
       board[i][0] = 1;
       board[0][i] = 1;
   }
   for( int i = 1; i < n; i++){
    for( int j = 1; j < n; j++){
        board[i][j] = board[i-1][j] + board[i][j-1];
    }
   }
   cout << board[9][9] << endl;
   
   for( int i = 0; i < n; i++){
    for( int j = 0; j < n; j++){
        cout << board[i][j] << " ";
    }
    cout <<endl;
   }
   
   for( int i = 1; i < n; i++){
    for( int j = 0; j < n; j++){
        if( j == 0 )
            cur[j] = 1; 
        else 
            cur[j] = cur[j-1] + prev[j];
    }
    prev = cur;
    cout << endl;
   } for( auto i = prev.begin(); i != prev.end(); ++i)
       cout << *i << " ";
   cout << endl;
    for( auto i = cur.begin(); i != cur.end(); ++i)
       cout << *i << " ";
    cout <<endl;
    cout << cur[n-1] << endl;
    cout << "done " << endl;
    std::cin.get();
    return 0;
}