Q. The maximum number of points of intersection of five lines and four circles is
Permutations and Combinations
Solution:
Two circles intersect at two distinct points.Two straight lines intersect at one point. One circle and one straight line intersect at two distinct points. Then the total numbers of points of intersections are as follows:
Number of ways of selection
Points of intersection
Two straight lines: $^5C_2$
$^5C_2 \times 1 = 10$
Two circles: $^4C_2$
$^4C_2 \times 2 = 12$
One line and one circle: $^5C_1 \times \,{}^4C_1$
$^5C_1 \times \,{}^4C_1 \times 2 = 40$
Total
62
| Number of ways of selection | Points of intersection |
|---|---|
| Two straight lines: $^5C_2$ | $^5C_2 \times 1 = 10$ |
| Two circles: $^4C_2$ | $^4C_2 \times 2 = 12$ |
| One line and one circle: $^5C_1 \times \,{}^4C_1$ | $^5C_1 \times \,{}^4C_1 \times 2 = 40$ |
| Total | 62 |