Problem 107 - Algebra for JEE ADVANCED

Question

The number of three-digit numbers of the form \(x y z\) such that \(x

Step-by-Step Solution

Answer

The total number of valid numbers is 285.

Step 1: Define the Problem

We need to count the number of three-digit numbers of the form \(x y z\) where \(x, y, z\) are digits, and they satisfy \(x < y\) and \(z \leq y\). Here, \(x\) is the hundreds digit, \(y\) is the tens digit, and \(z\) is the units digit.

Step 2: Determine Possible Values for x

The hundreds digit \(x\) can be any number from 1 to 9 (since \(x\) cannot be 0 as we want a three-digit number).

Step 3: Determine Values for y Given x < y

For each value of \(x\), \(y\) can be any digit greater than \(x\) and less than or equal to 9. This means for a specific \(x\), there are \(9-x\) possible choices for \(y\).

Step 4: Determine Values for z Given z <= y

For each combination of \(x\) and \(y\), the digit \(z\) can range from 0 to \(y\). This results in \((y + 1)\) choices for \(z\).

Step 5: Calculate the Total Number of Valid Numbers

We calculate the total number of valid combinations by iterating over possible values of \(x\) (1 through 9), and summing up the product of possible \(y\) and \(z\) values for each \(x\).For each \(x\), calculate the number of choices as \(\sum_{y=x+1}^{9} (y + 1)\).

Step 6: Perform the Summation

Calculate:- For \(x = 1\), \(y = 2, 3, ..., 9\): \(\sum_{y=2}^{9} (y + 1) = 3 + 4 + ... + 10 = 52\)- For \(x = 2\), \(y = 3, 4, ..., 9\): \(\sum_{y=3}^{9} (y + 1) = 4 + 5 + ... + 10 = 46\)- For \(x = 3\), \(y = 4, 5, ..., 9\): \(\sum_{y=4}^{9} (y + 1) = 5 + 6 + ... + 10 = 40\)- For \(x = 4\), \(y = 5, 6, ..., 9\): \(\sum_{y=5}^{9} (y + 1) = 6 + 7 + ... + 10 = 34\)- For \(x = 5\), \(y = 6, 7, ..., 9\): \(\sum_{y=6}^{9} (y + 1) = 7 + 8 +... + 10 = 28\)- For \(x = 6\), \(y = 7, 8, 9\): \(\sum_{y=7}^{9} (y + 1) = 8 + 9 + 10 = 24\)- For \(x = 7\), \(y = 8, 9\): \(\sum_{y=8}^{9} (y + 1) = 9 + 10 = 19\)- For \(x = 8\), \(y = 9\): \(\sum_{y=9}^{9} (y + 1) = 10 = 10\)- For \(x = 9\), \(y > 9\), no valid \(y\).Add these up: 52 + 46 + 40 + 34 + 28 + 24 + 19 + 10 = 285.

Key Concepts

Digit Arrangement

When it comes to forming numbers, the arrangement of digits is crucial. For example, in the problem of constructing a three-digit number such as \(xyz\), each of these letters represents different parts of a number.

• \(x\) is the digit in the hundreds place.
• \(y\) is the digit in the tens place.
• \(z\) is the digit in the units place.

These positions are not random; they influence the size and validity of the formed number. To arrange digits correctly:

• Ensure the hundreds place has a non-zero digit since we need a three-digit number.
• Tens and units digits can range from 0 to 9.

This concept is crucial in defining the number's value and its overall makeup, allowing us to determine eligibility according to specific mathematical conditions, such as inequalities.

Inequalities in Digits

In many problems involving number formation, there are rules or inequalities that the digits must satisfy. In our specific exercise, there are two main inequalities:

• \(x < y\): The digit in the hundreds place must be less than the digit in the tens place.
• \(z \leq y\): The digit in the units place must be less than or equal to the digit in the tens place.

These inequalities ensure that the numbers formed meet specific logical criteria. By following these conditions, we limit the possible combinations and make it easier to count the number of viable arrangements. Without these inequalities, we could have multiple numbers that don't adhere to such order, making our counting process neither restricted nor contextually meaningful.

Three-Digit Numbers

Three-digit numbers are exactly what they sound like - numbers that range from 100 to 999. Each digit plays an essential role in forming the entirety of the number:

• They have a three-part structure: hundreds, tens, and units place.
• Each digit can affect the magnitude and sequence of the number.

In our combinatorics problem, the aim is to construct these numbers using certain conditions (like inequalities) so that they abide by given rules. It is important to remember that the number must start with a non-zero digit, as starting with zero would create either a one- or two-digit number. This categorical definition helps organize problems and solutions, removing ambiguity surrounding the possible range and format of numbers.