<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 countIncreasingTeams(const vector<int>& rating, int currentIndex, int teamSize, vector<vector<int>>& increasingCache){
        int n = rating.size();
        if(teamSize == 3) return 1;

        if(increasingCache[currentIndex][teamSize] != -1){
            return increasingCache[currentIndex][teamSize];
        }

        int validTeams = 0;
        for(int nextIndex = currentIndex +1; nextIndex < n; nextIndex ++){
            if(rating[nextIndex] > rating[currentIndex]){
                validTeams += countIncreasingTeams(rating, nextIndex, teamSize +1, increasingCache);
            }
        }
        return increasingCache[currentIndex][teamSize] = validTeams;
    }
    int countDecreasingTeams(const vector<int>& rating, int currentIndex, int teamSize, vector<vector<int>>& decreasingCache){
        int n = rating.size();
        if(teamSize == 3) return 1;

        if(decreasingCache[currentIndex][teamSize] != -1){
            return decreasingCache[currentIndex][teamSize];
        }

        int validTeams = 0;
        for(int nextIndex = currentIndex +1; nextIndex < n; nextIndex ++){
            if(rating[nextIndex] < rating[currentIndex]){
                validTeams += countDecreasingTeams(rating, nextIndex, teamSize +1, decreasingCache);
            }
        }
        return decreasingCache[currentIndex][teamSize] = validTeams;
    }
        int numTeams(vector<int>& rating) {
        int n = rating.size();
        int teams = 0;
        vector<vector<int>> increasingCache(n, vector<int>(4,-1));
        vector<vector<int>> decreasingCache(n, vector<int>(4, -1));

        //calculate total teams by considering each soldier as a starting point
        for(int startIndex = 0; startIndex < n; startIndex++){
            teams += countIncreasingTeams(rating, startIndex, 1, increasingCache) +
                        countDecreasingTeams(rating, startIndex, 1, decreasingCache);
        }
        return teams;
    }
};
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