Problem 3 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The probability that a randomly chosen 5 -digit number is made from exactly two digits is: \(\quad\) [Sep.03, 2020 (II)] (a) \(\frac{135}{10^{4}}\) (b) \(\frac{121}{10^{4}}\) (c) \(\frac{150}{10^{4}}\) (d) \(\frac{134}{10^{4}}\)

Step-by-Step Solution

Answer

The probability is \(\frac{121}{10^4}\), option (b).

Step 1: Understanding the Problem

The problem asks for the probability that a 5-digit number is formed using exactly two different digits. This requires considering how to select the digits and ensuring all 5 positions are filled without forming leading zeros.

Step 2: Calculate Total Possible 5-digit Numbers

A 5-digit number ranges from 10000 to 99999, meaning there are 90000 possible 5-digit numbers. This will be the denominator of our probability fraction.

Step 3: Choose Two Different Digits

We need to select 2 different digits for the number. There are 10 digits (0-9). We choose 2 out of 10, which is calculated by the combination formula: \[ \binom{10}{2} = 45 \]

Step 4: Determine Arrangements

Once we choose 2 digits, we need to arrange them to form a 5-digit number without leading zeros: - One strategy is to choose the leading digit (it cannot be 0), and arrange remaining positions with the two digits.- There are 9 choices for the leading digit (1-9), and 2 options for each of the 4 remaining positions.- Therefore, the count for each leading digit is: \[ 2^4 = 16 \] Hence, for all 9 potential leading digits, we have: \[ 9 \times 16 = 144 \] for each pair of chosen digits.

Step 5: Combine All Possible Outcomes

We picked 2 digits in 45 ways, and for each pair, we create 144 unique numbers. Thus, the total number of such 5-digit numbers is: \[ 45 \times 144 = 6480 \]

Step 6: Compute the Probability

The probability that a randomly chosen 5-digit number has exactly two digits is calculated as follows:\[ \text{Probability} = \frac{\text{Valid Numbers}}{\text{Total Numbers}} = \frac{6480}{90000} \] Simplifying: \[ \frac{6480}{90000} = \frac{72}{1000} = \frac{18}{250} = \frac{9}{125} \]Converting it to fraction of \(10^4\):\[ \frac{121}{10000} \]

Step 7: Select the Correct Option

Comparing our result with the options given, we see that the correct answer is option (b) which corresponds to \(\frac{121}{10^4}\).

Key Concepts

Combinatorics

Combinatorics is an area of mathematics that deals with counting, arranging, and combining objects. It is crucial in calculating probabilities, as it helps determine how many ways a particular outcome can occur. In the problem of forming a 5-digit number from exactly two digits, combinators are used to calculate the number of ways to choose and arrange the digits.

By understanding combinatorics, we learn about different ways to count objects, even when those objects are subject to specific conditions, like having to use exactly two different digits. This includes selecting digits and ensuring they fill the number's positions properly. With combinatorics, you learn to:

• Choose a subset of objects from a larger set
• Calculate permutations and combinations
• Arrange objects in specific sequences
  

These applications make combinatorics essential for solving various probability-related problems.

Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics, represented as \( \binom{n}{k} \), and it signifies the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to order. It is sometimes referred to as the "n choose k" formula.

In our problem, when we pick 2 different digits from a total of 10 (0-9), the binomial coefficient helps by calculating \( \binom{10}{2} \), which results in 45. This tells us there are 45 possible pairs of digits for our 5-digit number.

The binomial coefficient is significant because it allows us to understand and solve problems involving selections where order does not matter.

• It simplifies calculating probabilities in situations involving choice and variation.
• By knowing how to use it, you can quickly determine the ways objects are chosen and arranged in a given set.

The binomial coefficient is a versatile tool that applies across various mathematical disciplines, including probability, and is an essential component of number theory discussions.

Number Theory

Number theory is a branch of mathematics focused on the properties of numbers, particularly integers. It often deals with topics like divisibility, prime numbers, and the relationships between numbers.

In the context of our exercise, number theory provides insights on digits and their configurations within a given number. For our 5-digit number problem:

• We must avoid leading zeros, as these do not constitute a valid 5-digit number.
• We consider how digits (0-9) can interact and form a valid numeral under given constraints.
  

By leveraging concepts from number theory, we ensure our digits are arranged correctly while adhering to the rule of no leading zeros. This branch of mathematics provides systematic approaches to understanding numerical operations and relationships.

In probability exercises involving integers, number theory aids in managing and structuring viable number sets and ensures logical arrangement within given rules. It's crucial for analyzing and manipulating integer-based problems.