<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 f(int indx, int count, vector<int>&rating, vector<vector<int>> &dp){
        if(count == 3)return 1;
        if(dp[indx][count]!=-1)return dp[indx][count];
        int res = 0 ;
        for(int i = indx+1; i < rating.size(); i++){
            if(rating[i] <= rating[indx]) continue;
            res += f(i, count+1, rating,dp);
        }
        return dp[indx][count] = res;
    }
    int fd(int idx, int cnt, vector<int> &rating, vector<vector<int>>&dp2) {
        // base cases
        if(cnt == 3) return 1;
        if(dp2[idx][cnt] != -1) return dp2[idx][cnt];

        int res = 0;
        for(int i=idx+1; i<rating.size(); ++i) {
            if(rating[i] >= rating[idx]) continue;
            res += fd(i, cnt+1, rating,dp2);
        }
        return dp2[idx][cnt] = res;
    }
    int numTeams(vector<int>& rating) {
        int ans = 0 ;
        int n = rating.size();
        // for(int i = 0 ; i < n ; i++){
        //     for(int j = i+1 ; j < n ; j++){
        //         for(int k = j+1; k < n; k++){
        //             if((rating[i] < rating[j] && rating[j] < rating[k]) || rating[i] > rating[j] && rating[j] > rating[k])ans++;
        //         }
        //     }
        // }
        vector<vector<int>> dp1(1000,vector<int>(3,-1));
        vector<vector<int>> dp2(1000,vector<int>(3,-1));
        int res = 0;
        for(int i=0; i<rating.size(); ++i) {
            res += f(i,1,rating,dp1);
            res += fd(i,1,rating,dp2);
        }
        return res ;
    }
};
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