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

Question

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

Step-by-Step Solution

Answer

There are 36 such two-digit numbers.

Step 1: Understand the Problem

We need to find all two-digit numbers where the second digit is smaller than the first. This means we have to determine the valid two-digit combinations of numbers.

Step 2: Define Constraints

For a two-digit number of the form \(xy\), where \(x\) is the tens digit and \(y\) is the units digit, we know the constraints: \(x\) can be any digit from 1 to 9, and \(y\) must be less than \(x\).

Step 3: Calculate Possible Numbers

For each possible value of \(x\), count the number of valid \(y\) values such that \(y < x\). Illustratively:- If \(x = 1\), then no possible value for \(y\) since \(y\) < 1.- If \(x = 2\), \(y\) values can be \([0, 1]\) which is 2 options.- If \(x = 3\), \(y\) values can be \([0, 1, 2]\) which is 3 options.- ... continue this until \(x = 9\), where \(y\) values can be \([0, 1, 2, 3, 4, 5, 6, 7, 8]\) which is 9 options.

Step 4: Summation of Valid Combinations

Sum the number of valid \(y\) digits for each \(x\) from 2 to 9:\[0 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 36\]

Step 5: Conclusion

We have calculated there are 36 valid two-digit numbers where the second digit is less than the first.

Key Concepts

Two-Digit Numbers

When we talk about two-digit numbers, we are referring to numbers from 10 to 99. These numbers are composed of two parts, or digits, which are placed next to each other in a specific sequence. In a two-digit number like \(xy\), \(x\) represents the tens digit, while \(y\) is the units digit.
The tens digit \(x\) can range from 1 to 9, because a zero in the tens place would not constitute a two-digit number. Similarly, the units digit \(y\) can range from 0 to 9, providing a broad range of possibilities for forming two-digit numbers.
These rules lay the foundation for understanding the constraints and possibilities in a problem involving two-digit numbers.

Digit Comparison

Digit comparison is the examination of how two digits relate to each other in a number. In this context, we focus on comparing the tens digit \(x\) and the units digit \(y\) of a two-digit number to see when one is larger than or less than the other.
For this specific exercise, we're interested in combinations where \(y < x\). This means the number on the right (units place) is smaller than the number on the left (tens place).
Understanding this relationship is crucial because it dictates the eligibility of numbers to be counted in the problem's solution. Practicing these comparisons helps to develop a deeper understanding of how digits can be systematically combined under given constraints.

Combinatorial Counting

Combinatorial counting involves finding the total number of ways something can occur, adhering to specific rules or constraints. In our problem of counting two-digit numbers where \(y < x\), this method helps us calculate systematically.
Consider each possible value for \(x\), from 1 to 9, and list the potential values of \(y\) where \(y < x\). Count these valid options for each \(x\). For example:

• If \(x = 2\), valid options for \(y\) are \(0, 1\).
• If \(x = 3\), valid options for \(y\) are \(0, 1, 2\).
• Continue this pattern up to \(x = 9\).

The key is to systematically add up all valid combinations, as done in the solution. This approach ensures that every possible pair is considered, and none are missed, resulting in the correct total number of eligible two-digit numbers.