In a college of 300 students every student reads 5 newspapers and every newspaper is read by 60 stud

Q: Wait, why do we equate the two totals? Shouldn't we be adding them?

Think of it like this: if you count the number of hands in a room by asking every person how many hands they have, you get the same number as if you count the total number of hands directly. We aren't adding the totals; we are looking at the same total from two different perspectives. The 'Double Counting Principle' is just a formal way of saying that the total number of 'connections' in a system is invariant, regardless of how you choose to count them.

Q: Does the order of students or newspapers matter?

Not at all! The beauty of this principle is that it is independent of the arrangement. Whether the students are sitting in rows or standing in a circle, the total number of reading events remains invariant. The math doesn't care about the physical layout; it only cares about the total number of connections.

Q: Is this problem just a simple division problem?

At its core, yes, but the JEE Advanced level isn't about the division; it's about recognizing the structure. Once you identify the 'Double Counting' pattern, the division is just the final step of a much deeper logical journey. Always look for these invariant quantities in combinatorics problems!

Question

In a college of 300 students every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspapers is

Options:

(A) at least 30

(B) at most 20

(C) exactly 25

(D) none of these

Sigma Percentile · Accepted Answer

### The Art of Counting: A Journey Through Combinatorics

Imagine you are standing in the heart of a bustling college campus. There are $300$ students, each with a thirst for knowledge, and a library filled with newspapers.

This isn't just a simple arithmetic problem; it is a classic exercise in the power of perspective. In the world of competitive mathematics, especially for JEE Advanced, the most complex problems often collapse into simplicity if you can just find the right angle to view them from.

Today, we are going to master the **Double Counting Principle**, a tool that will become one of the most reliable weapons in your problem-solving arsenal.

### The Student Perspective: The First View

Let us start by looking at the campus through the eyes of the students. We have $300$ students in total.

The problem tells us that every single student reads exactly $5$ newspapers. If we want to know the total number of 'reading events'—that is, every time a student picks up a newspaper—we simply multiply the number of students by the number of newspapers each one reads.

It is a straightforward calculation: 
$$300 	imes 5 = 1500$$

This number, $1500$, represents the total count of all reading interactions across the entire college. It is a fixed, immutable fact of this system.

### The Newspaper Perspective: The Second View

Now, let us shift our perspective. Instead of looking at the students, let us look at the newspapers. We don't know how many newspapers there are, so let us call this unknown quantity $n$.

The problem gives us another crucial piece of information: every single newspaper is read by exactly $60$ students. If we calculate the total number of reading events from this side, we have $n$ newspapers, and each one is read $60$ times.

Therefore, the total number of reading events must be $n 	imes 60$, or $60n$.

### The Bridge: The Double Counting Principle

Here is where the magic happens. We have calculated the exact same quantity—the total number of reading events—in two different ways.

From the student's perspective, it is $1500$. From the newspaper's perspective, it is $60n$. Because these two values represent the same physical reality, they must be equal.

This is the essence of the **Double Counting Principle**: 
$$60n = 1500$$

This equation is the bridge between the two perspectives. It allows us to solve for the unknown $n$ with absolute certainty.

### The Final Calculation

Now, we simply solve the equation: 
$$n = \frac{1500}{60}$$

By cancelling the zeros, we get: 
$$n = \frac{150}{6}$$

A quick division reveals that **$n = 25$**. Just like that, the mystery is solved. There are exactly $25$ newspapers in the library.

It is elegant, it is simple, and it is powerful. Remember, in your future exams, when you face a problem that seems to involve complex relationships between two sets, don't panic. Look for the invariant quantity, count it from both sides, and equate them. You will find that the most daunting problems often have the most beautiful, simple solutions.

Animated Solution for Mathematics - Sets and Relations: In a college of 300 students every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspapers is

Select Answer:

Visualized Solution

Identifying the Students

Identifying the Newspapers

The Double Counting Principle

Readings per Student

Total Readings (Student Perspective)

Readings per Newspaper

Total Readings (Newspaper Perspective)

Equating the Totals

Solving for $n$

Final Conclusion

The Sigma Insight: Types of Sets and Set Operations

The Art of Counting

The Student Perspective

The Newspaper Perspective

The Bridge

The Final Calculation

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