The maximum number of points of intersection of 10 circles will be:

Option c: 90. Solution: We are asked to find the maximum number of points of intersection for 10 circles. Let us analyze the problem: - Any two distinct circles can intersect at a maximum of 2 points. - To find the maximum number of intersection points for n circles, we first choose any 2 circles out of the n circles, and then multiply by the maximum number of intersection points per pair (which is 2 ). The number of ways to choose 2 circles out of n = 10 is given by 10C2 : 10C2 = 10!/2! × 8! = 10 × 9/2 × 1 = 45 pairs Since each pair of circles can intersect at a maximum of 2 points, the maximum number of points of intersection is: Maximum Points = 10C2 × 2 Maximum Points = 45 × 2 = 90 Generally, the formula for the maximum number of intersection points of n circles is: Max Points = n(n - 1) = 10 × 9 = 90 Hence, Option C is the correct answer.

The maximum number of points of intersection of $10$ circles will be:

The correct answer is Option c: $90$. We are asked to find the maximum number of points of intersection for $10$ circles. Let us analyze the problem: - Any two distinct circles can intersect at a **maximum of 2 points**. - To find the maximum number of intersection points for $n$ circles, we first choose any $2$ circles out of the $n$ circles, and then multiply by the maximum number of intersection points per pair (which is $2$). The number of ways to choose $2$ circles out of $n = 10$ is given by $^{10}C_{2}$: $$^{10}C_{2} = \frac{10!}{2! \times 8!} = \frac{10 \times 9}{2 \times 1} = 45 \text{ pairs}$$ Since each pair of circles can intersect at a maximum of $2$ points, the maximum number of points of intersection is: $$\text{Maximum Points} = ^{10}C_{2} \times 2$$ $$\text{Maximum Points} = 45 \times 2 = 90$$ Generally, the formula for the maximum number of intersection points of $n$ circles is: $$\text{Max Points} = n(n - 1) = 10 \times 9 = 90$$ Hence, **Option C** is the correct answer.

Permutations and CombinationsMCQMTP Dec 2023 Series IIQuestion 1740 of 251

All Questions

The maximum number of points of intersection of $\displaystyle 10$ circles will be:

Options

\displaystyle 2

\displaystyle 20

\displaystyle 90

\displaystyle 180

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option c — $\displaystyle 90$

All Options:

A $\displaystyle 2$
B $\displaystyle 20$
C $\displaystyle 90$
D $\displaystyle 180$

Detailed Solution & Explanation

We are asked to find the maximum number of points of intersection for

\displaystyle 10

circles.

Let us analyze the problem:
- Any two distinct circles can intersect at a **maximum of 2 points**.
- To find the maximum number of intersection points for

\displaystyle n

circles, we first choose any

\displaystyle 2

circles out of the

\displaystyle n

circles, and then multiply by the maximum number of intersection points per pair (which is

\displaystyle 2

).

The number of ways to choose

\displaystyle 2

circles out of

\displaystyle n = 10

is given by

\displaystyle ^{10}C_{2}

^{10}C_{2} = \frac{10!}{2! \times 8!} = \frac{10 \times 9}{2 \times 1} = 45 \text{ pairs}

Since each pair of circles can intersect at a maximum of

\displaystyle 2

points, the maximum number of points of intersection is:

\text{Maximum Points} = ^{10}C_{2} \times 2

\text{Maximum Points} = 45 \times 2 = 90

Generally, the formula for the maximum number of intersection points of

\displaystyle n

circles is:

\text{Max Points} = n(n - 1) = 10 \times 9 = 90

Hence, **Option C** is the correct answer.

About This Chapter: Permutations and Combinations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Factorials, Permutations, Combinations

This chapter deals with the fundamental principles of counting. It covers factorials, circular permutations, restricted permutations, combinations, and the differences between selecting items versus arranging them.

View Official ICAI Syllabus

Exam Strategy Tip

The most common mistake is confusing 'P' (Arrangement) with 'C' (Selection). If order matters (like opening a lock), use P. If order doesn't matter (like choosing a team), use C.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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The maximum number of points of intersection of 10\displaystyle 1010 circles will be: