First slide

Combinations

Question

The greatest number of points of intersection of n circles and m straight lines is

Moderate

A

$2 m n +^{m} C_{2}$
B

$\frac{1}{2} m (m - 1) + n (2 m + n - 1)$
C

$^{m} C_{2} + 2 (^{n} C_{2})$
D

none of these

Solution

The number of points of intersections
$\begin{array}{l} = (^{n} C_{2}) (2) + (^{m} C_{2}) (1) + (^{n} C_{1}) (^{m} C_{1}) (2) \\ = n (n - 1) + \frac{1}{2} m (m - 1) + 2 m n \end{array}$

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