<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>& rat) {
        int count = 0;
        vector<vector<int>>dpi(1000, vector<int>(3, -1));
        vector<vector<int>>dpd(1000, vector<int>(3, -1));
        int n = rat.size();
        // for (int i=0; i<n; i++)
        // {
        //     for (int j=i+1; j<n; j++)
        //     {
        //         if (rat[j] < rat[i])
        //             continue;
        //         for (int k=j+1; k<n; k++)
        //         {
        //             if (rat[k] > rat[j])
        //             {
        //                 count++;
        //               //  cout << rat[i] << " " << rat[j] << " " << rat[k] << endl;
        //             }
        //               //  count++;
        //         }
        //     }
        // }

        // for (int i=0; i<n; i++)
        // {
        //     for (int j=i+1; j<n; j++)
        //     {
        //         if (rat[j] > rat[i])
        //             continue;
        //         for (int k=j+1; k<n; k++)
        //         {
        //             if (rat[k] < rat[j])
        //             {
        //                 //cout << rat[i] << " " << rat[j] << " " << rat[k] << endl;
        //                 count++;
        //             }
        //                 //count++;
        //         }
        //     }
        // }

        for (int i=0; i<n; i++)
        {
            count += fi(i, 1, rat, dpi);
            count += fd(i, 1, rat, dpd);
        }
        return count;
    }

    int fi(int idx, int cnt, vector<int>&rating, vector<vector<int>>&dp)
    {
        if (cnt == 3)
            return 1;
        
        if (dp[idx][cnt] != -1)
            return dp[idx][cnt];
        
        int res = 0;

        for (int i = idx+1; i<rating.size(); i++)
        {
            if (rating[idx] >= rating[i])   continue;
            res += fi(i, cnt+1, rating, dp);
        }

        return dp[idx][cnt] = res;
    }

    int fd(int idx, int cnt, vector<int>&rating, vector<vector<int>>&dp)
    {
        if (cnt == 3)
            return 1;
        
        if (dp[idx][cnt] != -1)
            return dp[idx][cnt];
        
        int res = 0;

        for (int i = idx+1; i<rating.size(); i++)
        {
            if (rating[idx] <= rating[i])   continue;
            res += fd(i, cnt+1, rating, dp);
        }

        return dp[idx][cnt] = res;
    }

};
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