<p>Given an array of <code>points</code> where <code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> represents a point on the <strong>X-Y</strong> plane, return <em>the maximum number of points that lie on the same straight line</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/02/25/plane1.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> points = [[1,1],[2,2],[3,3]]
<strong>Output:</strong> 3
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/02/25/plane2.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> points = [[1,1],[3,2],[5,3],[4,1],[2,3],[1,4]]
<strong>Output:</strong> 4
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= points.length &lt;= 300</code></li>
	<li><code>points[i].length == 2</code></li>
	<li><code>-10<sup>4</sup> &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 10<sup>4</sup></code></li>
	<li>All the <code>points</code> are <strong>unique</strong>.</li>
</ul>

def points2rect(p0, p1):
    try:
        m = (p1[1] - p0[1])/(p1[0] - p0[0])
        q = p1[1] - m * p1[0]
    except:
        m = float(inf)
        q = p1[0]
    return round(m, 3), round(q, 3)

def fac(n):
    if n <= 1:
        return 1
    return n * fac(n-1)
    

def binom(n):
    return fac(n)/(fac(n-2)*2)

class Solution:
    def maxPoints(self, points: List[List[int]]) -> int:
        d = {}
        n = len(points)
        if n <= 2:
            return n
        r = 2
        for i, p0 in enumerate(points):
            for p1 in points[i+1:]:
                m, q = points2rect(p0, p1)
                if m not in d:
                    d[m] = {}
                d[m][q] = d[m].get(q, 0) + 1
                if d[m][q] > r:
                    r = d[m][q]
        for i in range(2, r+1):
            if binom(i) == r:
                return i
        return r
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