Question

Ten circles are all the same size. Each pair of these circles overlap but no circle is exactly on top of another circle. What is the greatest possible total number of intersection points of these ten circles?
A) 40
B) 70
C) 80
D) 90
E) 110

Answer 1

Hint:
Here we need to find the greatest possible number of intersection points of ten circles. For that, we will use the permutation to find the answer. First we will find the total number of intersecting points between two circles and then we will put the value of the number of intersecting points between two circles and the value of the number of circles in the formula of permutation. From there, we will get the value of the greatest possible number of intersection points of ten circles.

Complete step by step solution:
We need to find the greatest possible number of intersection points of ten circles.
Two circles can intersect in at most two points except the case in which two circles coincide with each other.
There are total 10 circles here.
So, by including all the possible pairs of circles, the maximum number of intersection points is given by
${}^{10}{{P}_{2}}$
We know the formula of permutation:
$\Rightarrow {}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$
On substituting the value of $n$ and b$r$ in this formula of permutation, we get
$\Rightarrow {}^{10}{{P}_{2}}=\dfrac{10!}{\left( 10-2 \right)!}$
On further simplification, we get
$\Rightarrow {}^{10}{{P}_{2}}=\dfrac{10!}{8!}$
On finding the factorial of 10 and 8, we get
$\Rightarrow {}^{10}{{P}_{2}}=\dfrac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}$
On simplifying the terms of numerator and denominator of this fraction, we get
$\Rightarrow {}^{10}{{P}_{2}}=10\times 9=90$
Thus, the total possible number of intersection points of ten circles is 90.

Hence, the correct option is option D.

Note:
Here we have calculated the value of terms like${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$, this is the ratio of factorial of the terms.
We need to know the meaning of the factorial for solving problems like that.
1) Factorial of any positive integer is defined as the multiplication of all the positive integers less than or equal to the given positive integers.
2) Factorial of zero is one.
3) Factorials are commonly used in problems of permutations and combinations.
4) Factorials of negative integers are not defined.