Problem 39 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of two digit numbers which are of the form \(x y\) with \(y

Step-by-Step Solution

Answer

The correct answer is 36, which is option D (None of these).

Step 1: Understanding the Problem

We have to find all two-digit numbers where the first digit, denoted by \( x \), is greater than the second digit, \( y \). In other words, \( y < x \).

Step 2: Identify Possible Digits for x

Since the number is a two-digit number, the possible digits for \( x \) range from 1 to 9. This means \( x \) can be 1, 2, 3, ..., or 9.

Step 3: Choose Suitable Digits for y Given x

For each digit \( x \), determine the valid digits for \( y \) such that \( y < x \). We will check each number from 1 to 9.

Step 4: Count Possible y for x=2

When \( x = 2 \), the possible values for \( y \) are 1. This gives 1 number: 21.

Step 5: Count Possible y for x=3

When \( x = 3 \), the possible values for \( y \) are 1 and 2, giving the numbers 31 and 32. This results in 2 numbers.

Step 6: Continue Counting for x=4 to x=9

- When \( x = 4 \), \( y \) can be 1, 2, 3 (3 numbers: 41, 42, 43).- When \( x = 5 \), \( y \) can be 1, 2, 3, 4 (4 numbers: 51, 52, 53, 54).- When \( x = 6 \), \( y \) can be 1, 2, 3, 4, 5 (5 numbers: 61, 62, 63, 64, 65).- When \( x = 7 \), \( y \) can be 1, 2, 3, 4, 5, 6 (6 numbers: 71, 72, 73, 74, 75, 76).- When \( x = 8 \), \( y \) can be 1, 2, 3, 4, 5, 6, 7 (7 numbers: 81, 82, 83, 84, 85, 86, 87).- When \( x = 9 \), \( y \) can be 1, 2, 3, 4, 5, 6, 7, 8 (8 numbers: 91, 92, 93, 94, 95, 96, 97, 98).

Step 7: Calculate Total Numbers

Add up all the numbers: 1 (for x = 2) + 2 (for x = 3) + 3 (for x = 4) + 4 (for x = 5) + 5 (for x = 6) + 6 (for x = 7) + 7 (for x = 8) + 8 (for x = 9) = 36.

Step 8: Conclude the Solution

The total number of two-digit numbers of the form \( xy \) with \( y < x \) is 36.

Key Concepts

Number Theory

Number theory is a fundamental branch of pure mathematics that deals primarily with the properties and behavior of numbers. In the context of this problem, understanding the system of digits and their relationships helps in dissecting two-digit numbers where the digits are integers and the position of each digit carries meaning. The two-digit number format, such as 21, can be viewed through the lens of place value. The first digit is in the 'tens' place, and the second is in the 'units' place. Such foundational ideas of number theory can further explore concepts like divisibility, modular arithmetic, and sequences, all of which rely on our understanding of these digits. In this specific exercise, understanding how one digit is larger than another within a number is crucial. It showcases the immediate arithmetic relation and inequality between numbers, illustrating how one component can affect another. This is a small steppingstone into larger number theory topics like prime numbers or greatest common divisors, which rely heavily on understanding how numbers interact with each other.

Integer Sequences

Integer sequences are sequences made up of whole numbers (positive or negative) and zero. Understanding integer sequences helps in recognizing patterns, predicting future entries, and resolving problems like the one in this exercise. In the exercise, we explore a series of numbers generated by adhering to a rule: the second digit is smaller than the first. Each possible value for \( x \) (1 through 9), when paired with a lesser value \( y \), creates a specific sequence.For example:

• For \( x = 2 \), we have the sequence: \( 21 \).
• For \( x = 3 \), the sequence expands with more elements: \( 31, 32 \).

Collectively, these sequences form a larger pattern as we count upwards, incrementally adding valid configurations of numbers. Recognizing such sequences builds our understanding of not just numbers, but how they can be meaningfully arranged to uncover insightful properties.

Problem-Solving Strategies

Problem-solving strategies are crucial for efficiently navigating exercises like the one provided. Breaking a problem down into smaller, more manageable parts is a key approach seen throughout the steps listed.One effective strategy is based on the concept of enumeration, which involves listing all possible options or solutions. By considering each possible starting digit for \( x \) and evaluating each subsequent \( y \), we can build a systematic approach to find all valid numbers. This method guarantees thoroughness and ensures no possibilities are overlooked.Another strategy shown here is the use of logical conditions, like the rule \( y < x \). By understanding and applying these constraints consistently, we effectively narrow our search to valid pairs.Moreover, summarizing and combining information step-by-step, as done in the solution, allows for clarity and simplification of complex tasks. These strategies are not just mathematical but applicable to various problem-solving situations involving constraints and sequential processes.