Problem 14 - A Complete Resource Book in Mathematics for JEE Main

Question

The sum of all the numbers that can be formed with the digits \(2,3,4,5\) taken all at a time is (A) 66666 (B) 84844 (C) 93324 (D) None of these

Step-by-Step Solution

Answer

The sum is 93324, option (C).

Step 1: Determine the Number of Permutations

Four digits can be arranged in different ways to form numbers. First, find the number of different permutations of the four digits 2, 3, 4, and 5. The formula for permutations is given by the factorial of the number of digits: \[ n! \]Here, \( n = 4 \), so the number of permutations is \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] Thus, there are 24 different numbers that can be formed.

Step 2: Calculate Contribution of Each Digit at Each Place Value

Each digit appears in each place (thousands, hundreds, tens, units) the same number of times in all permutations. Since there are 4 digits and 24 total numbers, each digit appears at each place: \[ \frac{24}{4} = 6 \] times.

Step 3: Find Total Contribution by Each Digit

Each position's value (thousands, hundreds, tens, units) will contribute to the sum of all numbers. Consider one position, say thousands, and calculate for repeat digits 6 times.The total contribution by each digit in one place is: \[ 6 \times 1000 \times (2 + 3 + 4 + 5) = 6 \times 1000 \times 14 = 84000 \] As this applies to each position, calculate the same for hundreds (100), tens (10), and units (1).

Step 4: Add the Contributions of All Places

The total contribution by each digit across all place values is calculated once and summed up for all places:- Thousands: 6 times the sum \[ = 6 \times 1000 \times (2 + 3 + 4 + 5) = 84000 \] - Hundreds: \[ = 6 \times 100 \times (2 + 3 + 4 + 5) = 8400 \] - Tens: \[ = 6 \times 10 \times (2 + 3 + 4 + 5) = 840 \] - Units: \[ = 6 \times 1 \times (2 + 3 + 4 + 5) = 84 \] Adding these up gives the total sum:\[ 84000 + 8400 + 840 + 84 = 93324 \]

Step 5: Determine the Correct Option

Compare your computed total sum with the given options. The result of 93324 matches Option (C).

Key Concepts

Permutations

When dealing with numbers formed by a set of digits, understanding permutations is key. A permutation is a specific arrangement of a set of items. Here, we are working with the digits 2, 3, 4, and 5.

• Permutations involve rearranging these digits to form unique sequences.
• With four digits, the permutations are represented mathematically by the factorial of the number of digits.

For a set of 4 digits, the number of permutations is equivalent to the factorial of 4, represented as \(4!\), which calculates to \(4 \times 3 \times 2 \times 1 = 24\). This means there are 24 unique numbers possible using all four digits.

Factorial

The factorial, denoted as \(n!\), is a fundamental concept in permutations and combinations. It represents the product of all positive integers from 1 to \(n\).

• Factorials are used to determine the number of possible arrangements or sequences for a set of items.
• For example, with 4 distinct digits, \(n = 4\), and the factorial \(4!\) helps to find all potential permutations.

The calculation \(4 \times 3 \times 2 \times 1\) results in 24, indicating there are 24 distinct sequences you can create with 4 different digits.

Place Value Contribution

Each digit contributes to the number it forms based on its position or place value. In a 4-digit number, from left to right, the positions are thousands, hundreds, tens, and units.

• The contribution of a digit depends on its place in a number. The further left the position, the larger the contribution.
• For example, a digit in the thousands place contributes much more to the number than one in the units spot.

During calculations, each digit appears equally in every position across all permutations. Hence, if a digit appears 6 times in every place, its overall contribution can be computed for each position and summed to find its total impact.

Digit Arrangement

Arranging digits to form numbers can be done by considering each digit's role in different positions. For a given set of digits, it is crucial to recognize their possible placements.

• Understanding the number of ways digits can be positioned helps calculate the overall number formed.
• In our example with digits 2, 3, 4, and 5, shifting their order creates new numbers, contributing to the total sum.
• Regardless of order, each digit takes turns occupying each position, ensuring equal representation across possible numbers.

This systematic approach guarantees that all configurations are considered, allowing for accurate computation of the total sum of permutations.