Introduction

In the realm of mathematical puzzles and problem-solving, logical reasoning and set theory play crucial roles. One such fascinating problem involves a class of 25 boys, each carrying a pen. The challenge lies in understanding the distribution of blue and red pens among them, leading to a deeper exploration of set theory and Venn diagrams. This article delves into the problem-solving strategy to find out how many boys carried pens of both colors.

The Scenario

Consider a classroom of 25 boys where every single boy possesses a pen. It is given that at least 20 boys have blue pens and 18 boys have red pens. The question is: how many boys carried pens of both colors?

Problem Analysis

Let's break down the problem step by step.

Step 1: Understanding the Givens

We are provided with the following data:

Total number of boys: 25 Number of boys with blue pens: 20 Number of boys with red pens: 18

Since there are 25 boys in total and at least 20 carry blue pens, this means that the remaining boys could potentially carry red pens. Similarly, the 18 boys with red pens could include some of the boys who also have blue pens.

Step 2: Application of Set Theory and Venn Diagrams

Let's use set theory and Venn diagrams to visualize and solve the problem. We can denote:

The set of boys with blue pens as Set B. The set of boys with red pens as Set R.

The intersection of these two sets, B ∩ R, represents boys who have both blue and red pens.

Step 3: Solving the Problem

Given that 20 boys have blue pens and 18 have red pens, we can use the principle of inclusion and exclusion:

N(B ∪ R) N(B) N(R) - N(B ∩ R)

where:

N(B ∪ R) is the total number of boys with at least one color pen (which is 25) N(B) 20 N(R) 18 N(B ∩ R) is the number of boys with both colors of pens

Substituting the values, we get:

25 20 18 - N(B ∩ R)

N(B ∩ R) 20 18 - 25

N(B ∩ R) 13

Therefore, there are 13 boys who have both blue and red pens.

Assumptions and Validations

An important assumption here is that no boy carries more than one pen of each color. If this assumption is violated, the solution might differ.

Conclusion

The problem showcases the power of set theory and logical reasoning in solving mathematical puzzles. By breaking down the given data and applying the principle of inclusion and exclusion, we can accurately determine the number of boys carrying pens of both colors. This type of problem-solving is not only intellectually stimulating but also reinforces fundamental mathematical concepts.