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There are 10 points in a plane out of which 4 points are collinear. How many straight lines can be drawn by joining any two of them?
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Verified Answer
The correct answer is:
40
From 10 given points, ${ }^{10} C_2$ straight lines can be drawn.
But 4 points are collinear, using 4 points, ${ }^4 C_2$ straight lines can be drawn.
From 4 collinear points, 1 straight line can be drawn.
So, total number of straight lines $={ }^{10} C_2-{ }^4 C_2+1$
$\begin{aligned}
& =\frac{10 !}{8 ! 2 !}-\frac{4 !}{2 ! 2 !}+1 \\
& =45-6+1=40
\end{aligned}$
But 4 points are collinear, using 4 points, ${ }^4 C_2$ straight lines can be drawn.
From 4 collinear points, 1 straight line can be drawn.
So, total number of straight lines $={ }^{10} C_2-{ }^4 C_2+1$
$\begin{aligned}
& =\frac{10 !}{8 ! 2 !}-\frac{4 !}{2 ! 2 !}+1 \\
& =45-6+1=40
\end{aligned}$
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