<p>There are <code>n</code> soldiers standing in a line. Each soldier is assigned a <strong>unique</strong> <code>rating</code> value.</p>

<p>You have to form a team of 3 soldiers amongst them under the following rules:</p>

<ul>
	<li>Choose 3 soldiers with index (<code>i</code>, <code>j</code>, <code>k</code>) with rating (<code>rating[i]</code>, <code>rating[j]</code>, <code>rating[k]</code>).</li>
	<li>A team is valid if: (<code>rating[i] &lt; rating[j] &lt; rating[k]</code>) or (<code>rating[i] &gt; rating[j] &gt; rating[k]</code>) where (<code>0 &lt;= i &lt; j &lt; k &lt; n</code>).</li>
</ul>

<p>Return the number of teams you can form given the conditions. (soldiers can be part of multiple teams).</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> rating = [2,5,3,4,1]
<strong>Output:</strong> 3
<strong>Explanation:</strong> We can form three teams given the conditions. (2,3,4), (5,4,1), (5,3,1). 
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> rating = [2,1,3]
<strong>Output:</strong> 0
<strong>Explanation:</strong> We can&#39;t form any team given the conditions.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> rating = [1,2,3,4]
<strong>Output:</strong> 4
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>n == rating.length</code></li>
	<li><code>3 &lt;= n &lt;= 1000</code></li>
	<li><code>1 &lt;= rating[i] &lt;= 10<sup>5</sup></code></li>
	<li>All the integers in <code>rating</code> are <strong>unique</strong>.</li>
</ul>

#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>

using namespace __gnu_pbds;
class Solution {
template<typename T>
using ordered_set = tree<T, null_type, std::less<T>, rb_tree_tag, tree_order_statistics_node_update>;
public:
    int numTeams(vector<int>& rating) {
        multiset<int> P;
        ordered_set<int> O;
        vector<int> prefix1;
        vector<int> prefix2;
        for (int i = 0; i < rating.size(); i++) {
            // prefix1.push_back(distance(P.begin(), P.lower_bound(rating[i])));
            // prefix2.push_back(distance(P.upper_bound(rating[i]), P.end()));
            // P.insert(rating[i]);
            prefix1.push_back(O.order_of_key(rating[i]));
            prefix2.push_back(O.size()-O.order_of_key(rating[i]));
            O.insert(rating[i]);
        }
        set<int> S;
        ordered_set<int> N;
        vector<int> suffix1(rating.size());
        vector<int> suffix2(rating.size());
        for (int i = rating.size()-1; i >= 0; i--) {
            // suffix1[i] = distance(S.upper_bound(rating[i]), S.end());
            // suffix2[i] = distance(S.begin(), S.lower_bound(rating[i]));
            // S.insert(rating[i]);
            suffix2[i] = N.order_of_key(rating[i]);
            suffix1[i] = N.size()-N.order_of_key(rating[i]);
            N.insert(rating[i]);
        }
        int count = 0;
        for (int i = 1; i < rating.size()-1; i++) {
            count += prefix1[i]*suffix1[i] + prefix2[i]*suffix2[i];
        }
        return count;
    }
};
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