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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?

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#### 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 = 11C_{2 }

= `(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

= ^{4}C_{2}

= `(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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