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  • If no 2 of the indefinite straight lines connecting 9 points in a plane are coincident or parallel, and no 6 travel through the same point, then none of the 9 points can be connected. (wi

If no 2 of the indefinite straight lines connecting 9 points in a plane are coincident or parallel, and no 6 travel through the same point, then none of the 9 points can be connected. (with the expectation of the original 9) . Find the number of distinct points of intersection is equal to

Option: 1

232
 


Option: 2

350


Option: 3

384

 


Option: 4

432


Answers (1)

If there are 9 points and each line segment has 4 points, the number of lines that may be formed from the 9 points is given by,

\mathrm{ { }^9 C_2=\frac{9 !}{2 ! 7 !} }

\mathrm{{ }^9 C_2=36 }
The point of intersection obtained from these lines is given by,
\mathrm{ { }^{36} C_2=\frac{36 !}{2 ! 34 !} }

\mathrm{ { }^{36} C_2=35 \times 18 }

\mathrm{ { }^{36} C_2=630 }

Now we count how many times the initial 9 points appear at the point of intersection. 

A1 is the point of intersection of any two of the remaining 8 lines, therefore let's say we attach A1 to the remaining 8 points to form 8 lines.

The number of times the original 9 points come in the intersection is given by,

\mathrm{\begin{aligned} &{ }^8 C_2=\frac{8 !}{2 ! 6 !}\\ &{ }^8 C_2=28 \end{aligned}}

Thus, for 9 points we get,

28\times 9=252

Also, the 9 original points will intersect at least one time. Hence, we should take the count of 9 points.

Therefore, the total number of required distinct points of intersection is 630 - 252 + 9 = 384.

 


 

Posted by

Kshitij

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