Problem 33 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Number greater than 1000 but less than 4000 is formed using the digits \(0,1,2,3,4\) (repetition allowed). Their number is (a) 125 (b) 105 (c) 374 (d) 625

Step-by-Step Solution

Answer

The correct answer is 375, but none of the options matches this number.

Step 1: Understand the number range

The problem asks for numbers greater than 1000 but less than 4000, using the digits 0, 1, 2, 3, and 4 with repetition allowed. This means we are considering four-digit numbers.

Step 2: Determine possible thousands digit

The first digit of the number must be between 1 and 3. This is because it must be greater than or equal to 1000 and less than 4000. Therefore, the choices for the first digit are 1, 2, or 3, giving us 3 possible options for the thousands place.

Step 3: Determine possible hundreds, tens, and units digits

Since repetition is allowed, each of the hundred, ten, and unit places can be filled with any of the 5 digits: 0, 1, 2, 3, or 4.

Step 4: Calculate total number of combinations

For each valid choice of the thousands digit, there are 5 options for the hundreds digit, 5 options for the tens digit, and 5 options for the units digit. Therefore, the total number of possible numbers is given by multiplying the number of options for each place: \[ 3 \times 5 \times 5 \times 5 = 375 \].

Key Concepts

Number Formation

Number formation involves creating numbers following specific rules or constraints. In this exercise, we're forming four-digit numbers. The range is set between 1000 and 4000. This means the numbers must have four digits. Number formation considers the place value of each digit.

• Thousands Place: The first digit determines the number's size range. Here, it restricts the digits to 1, 2, or 3 to keep the number within the specified range.
• Other Places: The hundreds, tens, and units places help build the complete number by adding values that don't affect the thousands place's primary range condition.

Number formation can involve various digits, but how you choose and arrange these digits is crucial to meeting problem constraints.

Repetition Allowed

In this context, 'repetition allowed' means that each digit can be used multiple times in forming the number. This rule expands the potential combinations available.

Understanding Repetition

Allowing repetition often simplifies the calculation process by providing more options for each digit place.

• Each digit can appear in more than one place within a number, enhancing the diversity of possible numbers.
• Repetition doesn't affect the legality of a number. A single digit like '2' could appear in all four places, or not at all, depending on how the number is formed.

Repetition increases the number of options exponentially, as seen in the example: with five digit options (0 through 4), each place has 5 choices regardless of what occupies any other place.

Digit Constraints

Digit constraints limit which digits can be placed in certain positions, impacting the structure and quantity of different numbers you can form. In our exercise, the constraints are driven by both range and available digits.

Range Constraints

The digit constraints pertain to the thousands place mainly. Numbers must be more than 1000 but less than 4000, restricting the thousands digit to 1, 2, or 3.

Digit Pool

We're limited to using the digits 0, 1, 2, 3, and 4. This constraint affects all places but chiefly the thousands place, as it ensures our number falls within the required range.

• The thousands digit has 3 options (1, 2, or 3), while the hundreds, tens, and units places each have 5 digits (0 through 4).
• Applying these constraints results in a more manageable set of number possibilities, narrowing down the vast number pool to a solvable problem size.

Understanding and applying such constraints is key to solving problems efficiently in combinatorics.