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Question
There are 11 points in a plane. No three of these lies in the same straight line except 4 points, which are collinear. Find, the number of straight lines that can be obtained from the pairs of these points?
Solution
Number of points in a plane = 11
No three of these points lie in the same straight line except 4 points.
The number of straight lines that can be obtained from the pairs of these points
Through any two points, we can draw a straight line.
∴ The number straight lines through any two points of the given 11 points = 11C2
= `(11!)/(2! xx (11 - 2)!)`
= `(11!)/(2! xx 9!)`
= `(11 xx 10 xx 9!)/(2! xx 9!)`
= `(11 xx 10)/(2 xx 1)`
= 11 × 5
= 55
Given that 4 points are collinear.
The number of straight lines through any two points of these 4 points is
= 4C2
= `(4!)/(2!(4 - 2)!)`
= `(4!)/(2! xx 2!)`
= `(4 xx 3xx2!)/(2! xx 2!)`
= `(4 xx 3)/(2 xx 1)`
= 2 × 3
= 6
Since these 4 points are collinear
These 4 points contribute only one line instead of the 6 lines.
The total number of straight lines that can be drawn through 11 points on a plane with 4 of the points being collinear is
= 55 – 6 + 1
= 50
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