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There are 13 points in a plane of which 5 are collinear. Find the number of straight lines obtained by joining these points in pairs.


1. 68
2. 69
3. 72
4. 74

1 Answer

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Best answer
Correct Answer - Option 2 : 69

Concept:

The number of ways to select r things out of n things is given by \(\rm ^nC_r\)

\(\rm ^nC_r=\frac{n!}{(n-r)!\times(r)!}=\frac{n\times(n-1)\times....(n-r+1)}{r!}\)

 

Calculation:

To form a line we have to select two points out of 13 points

∴Number of lines = \(\rm ^{13}C_2=\frac{13\times12}{2\times 1}\)

= 78

Also number of lines out of 5 points = \(\rm ^5C_2=\frac{5\times4}{2\times 1}\)

= 10

But, these 5 points are collinear, and only one line can be formed out of these points.

∴ The total number of straight lines obtained by joining these points in pairs

= 78 - 10 + 1           ....(We add 1, as one line can be obtained out of 5 collinear points).

= 69

Hence, option (2) is correct. 

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