Problem 52 - A Complete Resource Book in Mathematics for JEE Main 2017

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: Determine the Choice for Each Digit

The first digit of the 4-digit number can't be 0 since it's a 4-digit number. Thus, there are 9 possible choices for the first digit (1-9).

Step 2: Choose the Second Digit

For the second digit, we can use any digit except the one used as the first digit, allowing for 9 possible choices (including 0 now since the number won't start with 0).

Step 3: Choose the Third Digit

The third digit can be any digit except the two already used, giving us 8 options.

Step 4: Choose the Fourth Digit

The fourth digit can be any digit except the three already used, leaving us with 7 options.

Step 5: Calculate the Total Number of Distinct 4-Digit Numbers

The total number of 4-digit numbers with distinct digits is calculated by multiplying the number of choices for each digit: \(9 \times 9 \times 8 \times 7 = 4536\).

Key Concepts

Permutations

Permutations are a fundamental part of combinatorics, dealing with the arrangement of objects in a specific order. In the context of 4-digit numbers, permutations help calculate how many ways we can arrange a certain set of digits. For example, if we have a limited selection of numbers and each number must be unique in its position, permutations equip us with the tools to find out how many unique sequences we can create.

When thinking about permutations of four digits, you begin by selecting which digit will occupy each of the places, starting from the first and moving sequentially to the fourth. It is important in permutations to consider that each arrangement is unique based on the order of digits. It’s all about creating a sequence where order matters, as opposed to combinations where order is not considered.

In our case, calculating permutations for 4-digit numbers involves determining the number of different ways we can arrange the digits, keeping in mind that no digit repeat is allowed. This systematic method uses factorial notation and understanding the choice available for each position, ensuring each digit is distinct across the number.

Distinct digits

Distinct digits imply that each digit in the number must be different from the others. They're particularly important in problems where duplication might distort the results you're looking for, such as in our 4-digit number scenario.

To ensure all digits are distinct, careful selection is necessary at each step. The first digit provides numerous choices, but each subsequent digit restricts the available options since chosen digits cannot reappear in other places.

• For the first digit, in most situations, you’ll have nine options to choose from if a number cannot start with zero.
• Once you've chosen the first digit, the second digit's choice is restricted to the remaining unchosen digits, which now includes zero.
• As the selection continues to the third and fourth digits, available options progressively reduce because the newly chosen digit cannot be any that were already placed in the previous positions.

This necessity of distinctness leads to a multiplication of the shrinking set of choices, thereby revealing how distinct digits create a cascading effect on the choices available for forming a complete number.

4-digit numbers

When tasked with forming 4-digit numbers, the specifics of digit placement become crucial. By definition, a 4-digit number is any number from 1000 to 9999, meaning the number must start with a non-zero digit.

The choice for each subsequent digit impacts the possible configurations, particularly when applying additional conditions such as using distinct digits. For each digit of a 4-digit number:

• The first digit must be chosen from 1 to 9, ensuring the number is indeed 4 digits long. This selection uses up one option from the possible digits.
• Following digits (second, third, and fourth) progressively use the remaining available numbers, being mindful that they cannot repeat any digits used earlier.

The approach in crafting 4-digit numbers with distinct digits combines the logic of permutations with specific constraints. This method showcases how combinatorial principles function in real-world scenarios, giving an efficient way to count possibilities by systematically narrowing the choices available at each step, culminating in our derived solution of 4536 combinations.