<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 {
private:
    int n;
    int mem[1001][1001] = {0};
public:
    int numTeams(vector<int>& rating) {
        n = rating.size();
        int res = 0;
        for (int i = 0; i < n; i++)
            res += dp(rating, i, 1);
        reverse(rating.begin(), rating.end());
        memset(mem, 1001 * 1001, 0);
        for (int i = 0; i < n; i++)
            res += dp(rating, i, 1);
        return res;
    }

    int dp(vector<int>& rating, int index, int count) {
        int res = 0;
        if (count == 2) {
            for (int i = index + 1; i < n; i++) {
                if (rating[i] > rating[index]) {
                    res++;
                }
            }
            mem[index][count] = res;
        } else {
            for (int i = index + 1; i < n; i++) {
                if (rating[i] > rating[index]) {
                    if (mem[i][count]) {
                        res += mem[i][count];
                    } else {
                        res += dp(rating, i, count + 1);
                    }
                }
            }
        }
        return res;
    }
};
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