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In a plane there are 37 straight lines of which 13 pass through point $A$ and 11 pass through the point $B$. Moreover, no three lines (apart from the lines passing through $A$ and $B$ ) pass through same point and no two are parallel. What is the number of points of intersection of the straight lines?
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The correct answer is:
${ }^{37} C_2-{ }^{13} C_2-{ }^{11} C_2+2$
Number of selection of two lines $={ }^{37} C_2$
Number of selection of two lines from lines which are concurrent at point $A={ }^{13} C_2$
Number of selection of two lines from lines which are concurrent at point $B={ }^{11} C_2$
$\therefore$ Total number of points of intersection
$$
={ }^{37} C_2-{ }^{13} C_2-{ }^{11} C_2+2
$$
Number of selection of two lines from lines which are concurrent at point $A={ }^{13} C_2$
Number of selection of two lines from lines which are concurrent at point $B={ }^{11} C_2$
$\therefore$ Total number of points of intersection
$$
={ }^{37} C_2-{ }^{13} C_2-{ }^{11} C_2+2
$$
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