Problem 65 - A Complete Resource Book in Mathematics for JEE Main 2017

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 answer is (C) 93324.

Step 1: Identify Possible Numbers

Calculate the number of different numbers that can be formed using the digits 2, 3, 4, and 5. Since we are using all four digits at a time, we permute 4 digits:\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]Thus, there are 24 different numbers.

Step 2: Use Symmetry Property of Digits

In a balanced set such as this one, each digit will appear in each positional place (units, tens, hundreds, thousands) an equal number of times. Since we have 24 numbers and 4 digits, each digit will appear \( \frac{24}{4} = 6 \) times in each place value.

Step 3: Calculate the Sum of One Digit Placements

Each digit contributes equally to each place value. Hence, the sum of the digits is:\[ 2 + 3 + 4 + 5 = 14 \]Since each contributes 6 times to each position, the total contribution of digits to any single place value is:\[ 6 \times 14 = 84 \]

Step 4: Calculate Total Sum Using Place Values

Add the contributions of each place value considering the weights (units, tens, hundreds, thousands):\[ 84 \times (1000 + 100 + 10 + 1) = 84 \times 1111 \]Now, calculate this product:\[ 84 \times 1111 = 93324 \]

Step 5: Verify and Answer Selection

Verify that our total matches one of the answer choices. Since our answer (93324) matches option (C), that is the correct choice.

Key Concepts

Digit Sum

When dealing with permutations of numbers, it's important to understand how to calculate the digit sum. In this context, the digit sum refers to the sum of all individual digits used to form numbers. Here, the digits are 2, 3, 4, and 5. To find the digit sum, simply add these numbers together: \[2 + 3 + 4 + 5 = 14\] This sum is essential for calculations later on, as it helps determine how much each individual digit contributes to the final number. In permutation scenarios, each digit must contribute to different positional values like units, tens, hundreds, etc., and the digit sum forms the basis for these contributions. Remember, understanding the digit sum helps simplify part of the solution process, especially when considering symmetry properties of permutations.

Positional Value

In permutations, the positional value of digits plays a critical role. Positional values are essentially the weight or place a digit holds on its contribution to a value. In a four-digit number like those formed by permuting 2, 3, 4, and 5, the positional values are the units, tens, hundreds, and thousands places.

• Units place contributes 1x the digit.
• Tens place contributes 10x the digit.
• Hundreds place contributes 100x the digit.
• Thousands place contributes 1000x the digit.

In our case, since each digit appears 6 times in each position (as calculated with 24 permutations), we use symmetry to calculate the sum for one positional value, namely 84. Then, we multiply by the aggregated weights, which is 1111 (i.e., 1000 + 100 + 10 + 1), to find the whole sum: \[84 \times 1111 = 93324\] Understanding positional value allows one to insightfully and systematically approach problems involving permutations and their contributions to numerical value.

Factorials

When it comes to permutations, factorials help us determine the number of ways digits can be arranged. The factorial of a number, denoted by an exclamation mark, such as 4!, is the product of all positive integers up to that number. For four distinct digits like 2, 3, 4, and 5, the factorial calculation determines how many unique numbers can be constructed using all digits once and only once: \[4! = 4 \times 3 \times 2 \times 1 = 24\] This means there are 24 possible permutations or arrangements of the digits. Factorials are a fundamental concept in permutations and combinations, underpinning the methodology for calculating distinct arrangements in a set and understanding the scope of possible number formations. Mastery of factorials is crucial for tackling problems related to arrangements and permutations effectively.