Circles Intersection Basics
When considering the intersection of circles, two distinct circles can intersect each other at a maximum of 2 points. This is the key principle for finding the overall maximum intersections.
Calculating Maximum Points for 5 Circles
To find the maximum number of intersection points for 5 circles, we need to determine how many unique pairs of circles can be formed. Each pair can contribute a maximum of 2 intersection points.
We can use the combinations formula to find the number of ways to choose 2 circles from a set of 5. The formula is:
\(C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}\)
In this case, \(n=5\) (total number of circles) and \(k=2\) (number of circles in each pair).
Number of pairs = \(C(5, 2) = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4}{2 \times 1} = 10\).
So, there are 10 unique pairs of circles.
Since each of these 10 pairs can intersect at a maximum of 2 points, the total maximum number of intersection points is:
Total Points = (Number of Pairs) \(\times\) 2 = \(10 \times 2 = 20\).
Therefore, the maximum number of points of intersection for 5 circles is 20.
Similar Questions
- Three perfect dice \(D_1, D_2\) and \(D_3\) are rolled. Let \(x, y\) and \(z\) represent the numbers on \(D_1, D_2\) and \(D_3\) respectively. What is the number of possible outcomes such that \(x < y < z\)?
- A man has \(7\) relatives (\(4\) women and \(3\) men). His wife also has \(7\) relatives (\(3\) women and \(4\) men). In how many ways can they invite \(3\) women and \(3\) men so that \(3\) of them are man's relatives and \(3\) of them are his wife's relatives?
- A triangle \(PQR\) is such that \(3\) points lie on the side \(PQ\), \(4\) points on \(QR\) and \(5\) points on \(RP\) respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed ?
- How many four-digit natural numbers are there such that all of the digits are even?
- How many 4-digit numbers are there having all digits as odd?
- In how many ways can the letters of the word INDIA be permutated such that in each combination, vowels should occupy odd positions?
- The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many such arrangements are there?
- What is the maximum number of possible points of intersection of four straight lines and a circle (intersection is between lines as well as circle and lines)?
- What is the sum of all four digit numbers formed by using all digits \(0, 1, 4, 5\) without repetition of digits?
- How many numbers greater than 1000 can be formed using the digits 0, 1, 2 and 3 (repetition of digits is not allowed) ?
Important Questions from Permutation and Combination
On a chess board, in how many different ways can 6 consecutive squares be chosen on the diagonals along a straight path ?
There are 6 persons arranged in a row. Another person has to shake hands with 3 of them so that he should not shake hands with two consecutive persons. In how many distinct possible combinations can the handshakes take place ?
In a tournament of Chess having 150 entrants, a player is eliminated whenever he loses a match. It is given that no match results in a tie/draw. How many matches are played in the entire tournament?
The letters A, B, C, D and E are arranged in such a way that there are exactly two letters between A and E. How many such arrangements are possible?
There is a numeric lock which has a 3-digit PIN. The PIN contains digits 1 to 7. There is no repetition of digits. The digits in the PIN from left to right are in decreasing order. Any two digits in the PIN differ by at least 2. How many maximum attempts does one need to find out the PIN with certainty?
