Chapter 19: Problem 13
What is the sum of all the 4 digit numbers which can be formed with the digits \(1,2,3,4\) without repetition? (a) 15560 (b) 87660 (c) 45600 (d) 66660
Short Answer
Expert verified
Answer: 66660
Step by step solution
01
Determine the total number of possible 4-digit combinations
Since there are 4 digits (1, 2, 3, and 4) and no repetition is allowed, there are 4 choices for the first digit, 3 choices for the second digit, 2 choices for the third digit, and 1 choice for the last digit. So, the total number of possible 4-digit numbers is 4 × 3 × 2 × 1 = 24.
02
Calculating the sum of all 4-digit numbers with given digits
To find the sum of all the 4-digit numbers, we'll consider the contribution of each digit individually.
Each digit appears the same number of times in every place (units, tens, hundreds, and thousands). In other words, each digit occurs 6 times in each place (since there are 24 possible arrangements and a digit can occupy a given place exactly 6 times). Therefore, the sum of the numbers formed by each digit at each place is the same:
Sum at each place =digit × frequency = (1 + 2 + 3 + 4) × 6 = 10 × 6 = 60.
We can now calculate the total sum by adding the contributions at each place:
Total Sum = (Units + Tens + Hundreds + Thousands) × Sum at each place = (1 + 10 + 100 + 1000) × 60 = 1111 × 60 = 66660.
03
Matching the answer with the options
The total sum of all 4-digit numbers formed with the given digits is 66660, which matches option (d). Hence, the correct answer is (d) 66660.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combinatorics
Combinatorics is the branch of mathematics that deals with counting, arranging, and combination of elements within a set. In the given problem, combinatorics is used to understand how many different 4-digit numbers can be formed with the digits \(1, 2, 3, 4\) without repetition.
The concept of permutations, a key topic in combinatorics, plays an important role here because permutations deal with the arrangement of elements. Since we are arranging 4 distinct digits, the number of such arrangements or "permutations" is calculated by multiplying the number of choices available at each step.
The concept of permutations, a key topic in combinatorics, plays an important role here because permutations deal with the arrangement of elements. Since we are arranging 4 distinct digits, the number of such arrangements or "permutations" is calculated by multiplying the number of choices available at each step.
- The first digit has 4 available choices.
- Once a digit is used, the next digit has 3 choices.
- The third digit will have 2 choices after two digits are used.
- The last digit will have only 1 choice.
Number Formation
Number formation is about creating numbers based on certain rules or criteria, such as using specific digits without repetition. Our exercise is a prime example of number formation. With the set of digits \(1, 2, 3, 4\), we want to form 4-digit numbers.
The critical rule here is "without repetition," meaning once a digit is used in one position of the number, it cannot be used again. This ensures each number is unique in its formation. Start by choosing the digits for each position:
The critical rule here is "without repetition," meaning once a digit is used in one position of the number, it cannot be used again. This ensures each number is unique in its formation. Start by choosing the digits for each position:
- The thousands place has 4 potential digits.
- The hundreds place has 3 remaining digits after the thousands place is filled.
- Then, the tens place will be filled from the 2 remaining digits.
- Finally, the units place will be the last digit left.
Digit Sum
Digit sum refers to the process of adding the value of the digits in numbers to evaluate their contribution in number formation. In our exercise, every digit from \(1, 2, 3, 4\) appears an equal number of times in each place (units, tens, hundreds, thousands) across all permutations, simplifying the calculation of the total sum of all numbers formed.
Here's how it works:
Here's how it works:
- Since the total number of permutations is 24, and each digit can appear in any place, each digit appears \(\frac{24}{4} = 6\) times in every place.
- The sum contributed by each digit in any specific place is \((1 + 2 + 3 + 4) = 10\).
- Thus, the sum for each position (units, tens, hundreds, thousands) becomes \(10 \times 6 = 60\).
Factorials
Factorials are mathematical expressions that represent the product of all positive integers up to a given number. In combination and permutations, factorials help calculate the number of ways to arrange given elements. The factorial function is denoted by \(!\).
In the exercise context, to understand how 24 different 4-digit numbers can be formed with digits \(1, 2, 3, 4\), we use a factorial calculation:
In the exercise context, to understand how 24 different 4-digit numbers can be formed with digits \(1, 2, 3, 4\), we use a factorial calculation:
- For 4 digits, we apply the factorial \(4!\).
- This equals \(4 \times 3 \times 2 \times 1 = 24\).