Problem 72 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

There are two urns. Urn A has 3 distinct red balls and um B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is [2010] (a) 36 (b) 66 (c) 108 (d) 3

Step-by-Step Solution

Answer

(c) 108

Step 1: Understanding the Problem

We have two urns: Urn A with 3 distinct red balls and Urn B with 9 distinct blue balls. We need to transfer two balls from each urn to the other, and we need to find the total number of distinct ways to do this.

Step 2: Choosing Balls from Urn A

There are 3 distinct red balls in Urn A. We need to choose 2 out of these 3 balls to transfer to Urn B. The number of ways to choose 2 balls out of 3 is given by the combination formula \[ \binom{3}{2} = 3 \]

Step 3: Choosing Balls from Urn B

There are 9 distinct blue balls in Urn B. We need to choose 2 out of these 9 balls to transfer to Urn A. The number of ways to choose 2 balls out of 9 is given by \[ \binom{9}{2} = 36 \]

Step 4: Calculating the Total Number of Ways

Since we need to choose 2 balls from Urn A and 2 balls from Urn B simultaneously, we multiply the number of ways to choose the balls from each urn:\[ \text{Total number of ways} = 3 \times 36 = 108 \]

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that deals with counting, arranging, and finding patterns. It helps us solve problems related to choosing and arranging objects. In our problem, we used combinatorics to find out how many ways we can transfer balls between two urns. Combinatorics often involves:

• Permutations: This is about arranging objects in a specific order. It's like when you try different ways to line up books on a shelf.
• Combinations: Different from permutations, combinations involve selecting items without regard to order, which is what we did in our exercise.
• Pigeonhole Principle: A principle that helps prove that if you have more objects than containers, at least one container must hold more than one object.

The heart of combinatorics is finding efficient ways to count and understand arrangements and selections.

Combinations

Combinations are a key concept in combinatorics. They focus on selecting items from a larger set, without caring about their order. For example, choosing toppings for your pizza doesn't need an order, just like selecting balls from urns in our problem. The main formula used for combinations is:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]This formula calculates how many different groups of \(r\) items can be selected from \(n\) items. Here are some key points:

• "n" is the total number of items to choose from.
• "r" is the number of items to choose.
• The exclamation mark "!" represents factorial, which means multiplying a series of descending natural numbers.

In our example, we used \(\binom{3}{2} = 3\) to find how many ways we could choose 2 balls from 3, and \(\binom{9}{2} = 36\) for choosing from 9 balls. The order of choosing doesn’t matter, which is what makes it a combination.

Discrete Mathematics

Discrete mathematics involves studying mathematical structures that are distinct and separate values. Unlike continuous mathematics, which deals with smooth and continuous functions, discrete mathematics focuses on countable or distinct objects. Our exercise is a part of discrete mathematics as it involves selecting and counting distinct items, which are represented by distinct red and blue balls in the urns. Key areas in discrete mathematics include:

• Graph theory: Deals with networks of points connected by lines, used in computer science for data organization.
• Set theory: The foundation for various topics, dealing with the collection of objects.
• Probability: Although more broadly applied, discrete probability is crucial in understanding random processes over finite sets.

The ability to count distinct selections, like in our urns scenario, is what brings combinatorics and discrete mathematics together, ensuring we understand how to systematically approach counting problems.