Question

The average of the four digits numbers that can be formed by using each of the digits \(3,5,7\), and 9 exactly once in each number is (a) 4444 (b) 5555 (c) 6666 (d) 7777

Short answer

The average of the 4-digit numbers that can be formed using the digits \(3, 5, 7,\) and 9 exactly once in each number is 7000. None of the given options (a) 4444, (b) 5555, (c) 6666, or (d) 7777 is correct.

Step-by-step solution

Generate all possible 4-digit numbers

Given that we can use each of the digits 3, 5, 7, and 9 exactly once per number, there are 4! (4-factorial) different possibilities. Let's list them below: 3579, 3597, 3759, 3795, 3957, 3975, 5379, 5397, 5739, 5793, 5937, 5973, 7359, 7395, 7539, 7593, 7935, 7953, 9357, 9375, 9537, 9573, 9735, 9753

Calculate the sum of the numbers

Now, let's calculate the sum of all these numbers. This can be done easily using a calculator: $3579 + 3597 + 3759 + 3795 + 3957 + 3975 + 5379 + 5397 + 5739 + 5793 + 5937 + 5973 + 7359 + 7395 + 7539 + 7593 + 7935 + 7953 + 9357 + 9375 + 9537 + 9573 + 9735 + 9753 = 168000$

Calculate the average of the numbers

Finally, we have to find the average of these numbers. Since there are 24 (4!) numbers, we'll divide the sum by 24: \(\frac{168000}{24} = 7000\) The average of the 4-digit numbers that can be formed using the digits \(3, 5, 7,\) and 9 exactly once in each number is 7000. None of the given options (a) 4444, (b) 5555, (c) 6666, or (d) 7777 is correct.

Key concepts

Permutations and Combinations

Understanding the concept of permutations and combinations is crucial when dealing with problems involving the arrangement of numbers or objects. Permutations refer to the different ways in which a set of items can be arranged where the order is important. For example, the digits 3, 5, 7, and 9 can form various unique 4-digit numbers depending on the order in which they're arranged.

Combinations, on the other hand, are about selecting items from a group where the order doesn't matter. In our exercise, we are interested in permutations since the arrangement of digits affects the resultant number. With four distinct digits, the number of permutations is calculated using factorial, denoted as 4! (4 factorial), which equals 24. That's why there are 24 different 4-digit numbers possible from the digits 3, 5, 7, and 9.

Understanding permutations is not only key to solving this problem but also a foundational element in many fields including mathematics, computer science, and statistics. It helps us quantify the possibilities in a scenario where order plays a significant role.

Mathematical Averages

The concept of a mathematical average, also known as the mean, is a fundamental statistic that shows the central tendency of a set of numbers. It is found by adding all the numbers together and then dividing by the count of those numbers. In our problem, we are looking for the average of all possible 4-digit numbers formed by 3, 5, 7, and 9.

This involves first calculating the sum of all the permutations of these digits and then dividing by the number of permutations. The average gives us an estimation of where the middle of the data set lies. In practical terms, it helps in various real-world situations, such as understanding average scores, economic data, or any other field where understanding the typical value is important.

A key aspect of calculating averages is accuracy in adding all the numbers and dividing by the correct count. The given solution made an error in the average calculation, illustrating the importance of careful computation and verification.

Factorial

The factorial of a non-negative integer is the product of all positive integers less than or equal to that number, which is fundamental in the study of permutations and combinations. The notation 'n!' represents the factorial of 'n'. For example, the factorial of 4 (4!) is calculated as 4 x 3 x 2 x 1 which equals 24.

Factorials are not just theoretical constructs; they appear frequently in different areas such as in calculating probabilities, arranging objects, or in algorithm complexities in computer science as they give us the count of all possible permutations of a certain number of objects.

In our problem, we used 4! to determine that there are 24 different ways to arrange the digits 3, 5, 7, and 9. A strong grasp of factorials is a powerful tool for solving many mathematical problems that involve counting and arranging distinct elements.