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  • There exist 27 coplanar, unparallel straight lines, of which 10 pass through a point and another 11 pass through the other point such that no line ever passes through both of these two points. No o

There exist 27 coplanar, unparallel straight lines, of which 10 pass through a point and another 11 pass through the other point such that no line ever passes through both of these two points. No other point has even three lines through it. Find the number of points of intersection among the straight lines.

Option: 1

250


Option: 2

251


Option: 3

253


Option: 4

Cannot be determined


Answers (1)

best_answer

Note the following:

  • The formula for the combination for the selection of the \mathrm{x} items from the \mathrm{y}different items is \mathrm{=^{y}C_{x}=\frac{y!}{x!(y-x)!}}

  • Two non-parallel straight lines result in a point of intersection. 

  •  The number of points of intersection of 27 coplanar, unparallel straight lines is \mathrm{=^{27}C_{2}}

  •  Again 10 straight lines passing through the same point and 11 straight lines passing through another point gives the number of intersection points of the lines as

 

As per the available data, the following is clear.

 

\begin{aligned} & ={ }^{27} C_2-{ }^{10} C_2-{ }^{11} C_2+2 \\ & =\frac{27 !}{2 !(27-2) !}-\frac{10 !}{2 !(10-2) !}-\frac{11 !}{2 !(11-2) !}+2 \\ & =\frac{27 \times 26}{2}-\frac{10 \times 9}{2}-\frac{11 \times 10}{2}+2 \\ & =351-45-55+2 \\ & =253 \end{aligned}

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Anam Khan

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