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  4. Statement-1: The maximum number of points of intersection of 8 circles of unequal radii is 56. Statement-2: The maximum number of points into which 4 circles of unequal radii and 4 non coincident straight lines intersect, is 50.

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Q. Statement-1 : The maximum number of points of intersection of 8 circles of unequal radii is 56.
Statement-2 : The maximum number of points into which 4 circles of unequal radii and 4 non coincident straight lines intersect, is 50.

Permutations and Combinations

Solution:

Two circles intersect in 2 points.
$\therefore $ Maximum number of points of intersection
$= 2 ×$ number of selections of two circles from 8 circles
$= 2 × ^8C_2 = 2 × 28 = 56$
Statement $2 : 4$ lines intersect each other in $^4C_2 = 6$ points
4 circles intersect each other in $2 × ^4C_2 = 12$ points.
Further, one lines and one circle intersect in two points. So 4 lines will intersect four circles in 32 points.
Maximum number of points $= 6 + 12 + 32 = 50$