Problem 13 - Algebra for JEE ADVANCED

Question

Numbers greater than 1000 but not greater than 4000 , which can be formed with the digits \(0,1,2,3,4\) (repetition of digits is allowed), are a. 350 b. 375 c. 450 d. 576

Step-by-Step Solution

Answer

The number of such numbers is 375, option b.

Step 1: Understanding the Range of Numbers

The numbers must be greater than 1000 but not greater than 4000. Thus, we are dealing with four-digit numbers.

Step 2: Selecting the Thousands Digit

The first digit (thousands place) must be one of 1, 2, or 3 to ensure the number is greater than 1000 and not greater than 3999.

Step 3: Selecting the Remaining Digits

For the hundreds, tens, and units places, each digit can be any of 0, 1, 2, 3, or 4. Since repetition is allowed, each of these positions has 5 possibilities.

Step 4: Calculating Total Combinations

Calculate the total number of combinations using the formula: \[ 3 \times 5 \times 5 \times 5 = 375 \] The first digit has 3 choices (1, 2, or 3), and each of the remaining three digits has 5 choices.

Key Concepts

Permutations

Permutations refer to the different ways you can arrange a set of items. In the context of number formation, permutations help us find out how many distinct numbers we can create by arranging given digits. When we allow repetition of digits, this means some digits can be reused in these permutations, which can increase the total possible formations.
When considering permutations with specific constraints, such as smallest or largest values, we are often dealing with conditional permutations. In this problem, since we are only interested in numbers greater than 1000 but not greater than 4000, our selection of permutations must start with certain digits (1, 2, or 3) for the thousands place. Each arrangement of the remaining digits gives us a unique permutation.

Number Formation

Number formation is the process of arranging digits to form numbers. The problem specifies forming four-digit numbers, a situation that greatly benefits from understanding permutations to calculate how many numbers can be created.
The key to effective number formation lies in understanding place value. The thousands place dictates the range and size of the number. This is because the leftmost digit must make the number over 1000, yet stay under 4000. In this case, we choose 1, 2, or 3 for the thousands digit. The remaining digits only need to fit within the base 5 (digits 0-4). The mechanics of number placement and how each digit contributes to the final outcome are central to solving these kinds of problems.

Digit Repetition

Digit repetition allows one to use the same digit in multiple positions. This significantly enlarges the set of potential permutations.
Imagine forming a number where any sequence of digits can repeat. Here, aside from the thousands digit, the hundreds, tens, and units places can freely cycle through 0, 1, 2, 3, and 4. Since each of these three positions can choose from 5 possibilities independently, repetition permits the mathematical calculation of total permutations as a product of possibilities per digit.

Range Selection

Range selection involves choosing numbers within a specified boundary. For this exercise, we're picking numbers greater than 1000 but not exceeding 4000.
This constraint directly limits our viable candidates, particularly affecting the thousands digit. Our choice for this digit is restricted to those that align with the problem's range (1, 2, or 3). For other digits, selection is less restrictive, allowing the widest choice of all provided digits. Understanding these range limits is essential for accurate number formation and ensures that all constructed numbers meet the exercise's criteria.