<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:
    def numTeams(self, rating: List[int]) -> int:
        n=len(rating)
        dp_right={x:0 for x in rating}
        dp_left={x:0 for x in rating}
        nums=rating[:]
        def merge(l,mid,h):
            for i in range(l,mid+1):
                j=mid+1
                while j<=h and nums[j]<nums[i]:
                    j+=1
                dp_right[nums[i]]+=j-(mid+1)
            for j in range(mid+1,h+1):
                i=l
                while i<=mid and nums[i]<nums[j]:
                    i+=1
                dp_left[nums[j]]+=i-l
            i,j=l,mid+1
            temp=[]
            while i<=mid and j<=h:
                if nums[i]<nums[j]:
                    temp.append(nums[i])
                    i+=1
                else:
                    temp.append(nums[j])
                    j+=1
            while i<=mid:
                temp.append(nums[i])
                i+=1
            while j<=h:
                temp.append(nums[j])
                j+=1
            nums[l:h+1]=temp[:]
        def merge_sort(l,r):
            if l<r:
                mid=(l+r)//2
                merge_sort(l,mid)
                merge_sort(mid+1,r)
                merge(l,mid,r)
        merge_sort(0,n-1)
        res=0
        for i,x in enumerate(rating):
            if i==0 or i==n-1:continue
            left_small=dp_left[x]
            left_big=i-dp_left[x]
            right_small=dp_right[x]
            right_big=n-i-dp_right[x]-1
            res+=left_small*right_big
            res+=left_big*right_small
        return res
17700