Problem 52 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of 4 -digit numbers with distinct digits is (A) 504 (B) 4536 (C) 4634 (D) 5040

Step-by-Step Solution

Answer

The answer is (B) 4536.

Step 1: Understanding the Problem

We are asked to find the number of 4-digit numbers where all the digits are distinct. A 4-digit number must have the thousands digit between 1 and 9, and the other digits can be any integer between 0 and 9, as long as they are not repeated.

Step 2: Choosing the Thousands Digit

For the thousands place, we cannot have zero as a digit because it would not be a 4-digit number. Therefore, there are 9 possible choices (1 through 9) for the thousands digit.

Step 3: Choosing the Hundreds Digit

For the hundreds place, we have 9 choices left. We already used one digit for the thousands place, but now we include zero as a possibility.

Step 4: Choosing the Tens Digit

For the tens place, we need to choose a digit different from the ones used in the thousands and hundreds places, so we have 8 choices left.

Step 5: Choosing the Units Digit

For the units place, we need to choose a digit different from the ones already used, leaving us with 7 choices.

Step 6: Calculating Total Numbers

To find the total number of different 4-digit numbers, we multiply all the possibilities: \(9 \times 9 \times 8 \times 7 = 4536\).

Key Concepts

Distinct Digits

When we talk about distinct digits in the context of creating numbers, we refer to the requirement that each digit must be different from one another. This means no repetition of numbers throughout the entire number.
For example, a four-digit number like 1223 is not considered to have distinct digits because '2' is repeated. Each digit from thousands to units needs to be unique.
This concept is crucial in problems involving permutations, as it restricts the possible choices for each digit once a previous digit is chosen. Ensuring each choice is a fresh selection without repetition changes the total possible outcomes in a considerable way.

Four-Digit Numbers

A four-digit number is made up of digits from 0 to 9, but there are specific rules to follow. The first digit, called the 'thousands' digit, cannot be zero. This ensures the number actually has four digits.
For example, a number like 0325 only has three digits because it starts with zero and isn't a valid four-digit number. When constructing a four-digit number, one must choose this digit carefully.
Once the thousands digit is chosen among 1 to 9, the remaining places: hundreds, tens, and units, can use any digits from 0 to 9, but always sticking to the rule of distinct digits to avoid repetition.

• Thousands place: 9 possibilities (1-9)
• Hundreds place: 9 possibilities (including 0)
• Tens place: 8 possibilities
• Units place: 7 possibilities

This sequence of selections reflects the constraints of the problem, requiring each digit selection to adapt based on earlier choices.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of different elements in sets. It provides the tools needed to solve problems involving permutations and combinations.
In this problem, we use permutations to count how many four-digit numbers can be formed using distinct digits. Permutations are arrangements where order matters. That's why we carefully choose each digit step-by-step, ensuring all are distinct.
We start with 9 choices for the thousands digit, followed by different choices for the subsequent hundreds, tens, and units places, respecting the previously chosen digits' exclusion from selection.
The final result is calculated through multiplication of available choices: \(9 \times 9 \times 8 \times 7 = 4536\). This process reflects the fundamental principle of multiplication in combinatorics, which states that if there are n ways to do one thing, and m ways to do another, then there are \(n \times m\) ways to do both.