<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>

class Solution {
public:
    int numTeams(vector<int>& rating) {
        int n = rating.size();
        if (n < 3) return 0;
        
        vector<pair<int, int>> soldiers;
        for (int i = 0; i < n; i++) {
            soldiers.push_back({rating[i], i});
        }
        sort(soldiers.begin(), soldiers.end());
        
        vector<int> indexMap(n);
        for (int i = 0; i < n; i++) {
            indexMap[soldiers[i].second] = i;
        }
        
        return countTeams(indexMap, n, true) + countTeams(indexMap, n, false);
    }
    
private:
    int countTeams(const vector<int>& indexMap, int n, bool ascending) {
        vector<int> bit(n + 1, 0);
        int teams = 0;
        
        for (int i = 0; i < n; i++) {
            int rank = indexMap[i] + 1;
            int left = ascending ? getSum(bit, rank - 1) : getSum(bit, n) - getSum(bit, rank);
            int right = ascending ? n - rank - (getSum(bit, n) - getSum(bit, rank)) : rank - 1 - getSum(bit, rank - 1);
            teams += left * right;
            update(bit, rank, 1);
        }
        
        return teams;
    }
    
    void update(vector<int>& bit, int index, int val) {
        while (index < bit.size()) {
            bit[index] += val;
            index += index & (-index);
        }
    }
    
    int getSum(const vector<int>& bit, int index) {
        int sum = 0;
        while (index > 0) {
            sum += bit[index];
            index -= index & (-index);
        }
        return sum;
    }
};
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