Problem 93 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The sum of all numbers greater than 1000 formed by using the digits \(0,1,2,3\), no digit being repeated in any number, is (A) 38664 (B) 48664 (C) 58664 (D) None of these

Step-by-Step Solution

Answer

The sum is 29892, which does not match any given option.

Step 1: Understanding the Problem

We are required to find the sum of all numbers greater than 1000 that can be formed using the digits 0, 1, 2, and 3, without repetition of digits. This means every number formed should be a 4-digit number since the alternatives greater than 1000 are 4 digits long.

Step 2: Identifying Possible Numbers

The total numbers that can be formed from four digits (0, 1, 2, 3) where the leading digit is not zero are calculated by considering each digit as the Thousand place digit except 0, i.e., either 1, 2 or 3.

Step 3: Calculate Numbers for Each Starting Digit

For thousands place using digit 1, there are 3 permutations: 1203, 1230, 1302, 1320. For thousands place using digit 2, there are 3 permutations: 2013, 2031, 2103, 2130. For thousands place using digit 3, there are 3 permutations: 3012, 3021, 3102, 3120.

Step 4: Summing Each Set of Numbers

Now, we compute the sum for each group having a different starting digit.Sum for thousands place digit 1: \( 1203 + 1230 + 1302 + 1320 = 5055 \)Sum for thousands place digit 2: \( 2013 + 2031 + 2103 + 2130 = 8277 \)Sum for thousands place digit 3: \( 3012 + 3021 + 3102 + 3120 = 16560 \)

Step 5: Final Calculation: Total Sum of All Numbers

Add up the sums from all groups: \( 5055 + 8277 + 16560 = 29892 \).As per the choices provided, the sum should be compared; hence correcting calculation errors or selection if it differs, but there is no exact match here.

Key Concepts

Understanding Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and combination of objects. It's essential in solving problems involving permutations and combinations, like the one stated here. In this exercise, we deal with permutations of four digits: 0, 1, 2, and 3. A permutation involves arranging these digits in different sequences.

• When creating permutations, the order of digits matters.
• We need to consider all unique sequences of the digits without repetition.
• This leads to various possible 4-digit numbers, formed by every possible order of the digits.

Throughout this process, combinatorics helps us systematically count how many valid sequences or numbers can be formed. By identifying how these numbers are arranged, we solve complex problems by simplifying them into manageable parts.

Calculating the Digit Sum

In this problem, calculating the digit sum means adding together all numbers that can be formed with the given digits. This involves several steps:

• Determine valid numbers by iterating all permutations.
• Calculate the sum for each group of numbers starting with the same digit.
• Add the sums of these groups to find the total sum.

For each digit acting as the leading digit, we compute all possible combinations of the other digits. As demonstrated in the step-by-step solution, we split and group permutations by thousands place. For instance, with '1' as the leading digit, the valid numbers formed are "1203", "1230", "1302", and "1320".
These sums, once added, collectively contribute to our final solution, helping to verify or correct calculated results.

Exploring 4-Digit Numbers

The focus on creating 4-digit numbers stems from the constraint of the problem: forming numbers greater than 1000. By using permutations of digits 0, 1, 2, and 3, we ensure all numbers meet the requirement by having four places, including thousands, hundreds, tens, and units.

• Each number must start with a digit from 1 to 3, since 0 cannot make a valid 4-digit number.
• Each number is constructed once digits are placed in each positional value.
• This accounts for all possible valid numbers that can be derived within given constraints.

The approach of analyzing 4-digit numbers is foundational in problems involving permutations. Understanding this ensures clarity in determining precise solutions and validates the structured procedure we followed in our calculations.