<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:
    vector<int> tree;
    int n;
    int numTeams(vector<int>& rating) {
        vector<pair<int, int>> ret = countSmaller(rating);
        int ans = 0;
        for(int i=1;i<n-1;i++){
            int leftSmaller = ret[i].second;
            int leftBigger = i - leftSmaller;
            int rightSmaller = ret[i].first;
            int rightBigger = n - i - 1 - rightSmaller;

            ans += leftSmaller*rightBigger + leftBigger*rightSmaller;
        }
        return ans;
    }
    void update(int ind, int val){
        ind++;
        while(ind <= n){
            tree[ind] += val;
            ind += (ind & -ind);
        }
    }

    int query(int ind){
        ind++;
        int tot = 0;
        while(ind > 0){
            tot += tree[ind];
            ind -= (ind & -ind);
        }
        return tot;
    }
    vector<pair<int, int>> countSmaller(vector<int>& nums) {
        n = nums.size();
        tree.resize(n+1, 0);
        vector<pair<int, int>> vec;
        for(int i=0;i<nums.size();i++){
            vec.push_back({nums[i], i});
        }
        sort(vec.begin(), vec.end());
        vector<pair<int, int>> ret(n);
        for(int i=0;i<n;i++){
            int ind = vec[i].second;
            int y = query(n-1);
            int x = query(ind);
            ret[ind] = {y-x, x};
            update(ind, 1);
        }

        return ret;
    }
};
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