Question

Find the sum of all four-digit numbers that can be obtained by using the digits 1,2,3 , \(4,\) and \(5 .\) Repeats are not allowed. Explain your reasoning.

Short answer

The sum of all four-digit numbers is 499,950.

Step-by-step solution

Determine Total Number of Four-Digit Combinations

Given five digits (1, 2, 3, 4, and 5), the number of four-digit numbers we can form without repeating any digits is determined by selecting any four digits out of five and arranging them. This is calculated as the number of permutations of 5 taken 4 at a time, given by \( P(5,4) = \frac{5!}{(5-4)!} = 5! = 120 \) combinations, since \(5-4 = 1\).

Calculate Contribution of Each Digit per Position

Each digit will appear in each position (thousands, hundreds, tens, and units) the same number of times. Given 120 total numbers and 4 digits per position, each digit appears \( \frac{120}{4} = 30 \) times in each position.

Calculate Value Contribution by Position

Each position has a specific multiplier: thousands (1000), hundreds (100), tens (10), and units (1). If a digit appears 30 times in any position, its contribution to that position across all combinations is its value times 30 times the position multiplier. So, for any digit \(d\): contribution = \(d \times 30 \times \text{multiplier}\).

Calculate Total Sum for All Digits

Each digit from 1 to 5 contributes equally. Thus, we calculate contribution for each: \( (1+2+3+4+5) \times 30 = 15 \times 30 = 450 \). Now, apply this to all positions: - Thousands: 450 contribution \(\times 1000 = 450,000\)- Hundreds: 450 contribution \(\times 100 = 45,000\)- Tens: 450 contribution \(\times 10 = 4,500\)- Units: 450 contribution \(\times 1 = 450\)Total sum = 450,000 + 45,000 + 4,500 + 450 = 499,950.

Key concepts

Permutations

Permutations are a fundamental concept in combinatorics, which involves the arrangement of items in a specific order. When dealing with permutations, the key point to remember is that the order of the elements matters. In this exercise, we are arranging the digits 1, 2, 3, 4, and 5 to create four-digit numbers. Since each number needs to be unique and use four different digits each time, we are specifically looking for permutations of 5 items taken 4 at a time. The number of such permutations can be found using the formula for permutations, which is represented as:\[ P(n, r) = \frac{n!}{(n-r)!} \]where \( n \) is the total number of items to choose from, and \( r \) is the number of items to arrange. For our example, we calculate \( P(5, 4) = \frac{5!}{(5-4)!} = 5! = 120 \). This result tells us that there are 120 different ways to arrange these four digits from a set of five.

Digit Arrangement

Digit arrangement is the core of creating different permutations in the exercise. When arranging digits, we start by selecting the specific digits we need and then determine their positions. For the given problem, we select 4 digits from the set {1, 2, 3, 4, 5} and then arrange them into positions of thousands, hundreds, tens, and units to form a four-digit number. Each digit can appear in any of these places, and the challenge is to figure out how many times each digit will appear in each position when all permutations are considered. Since there are 120 permutations, and each number uses 4 digits, it means each digit appears equally in every position. Thus, each digit will appear \( \frac{120}{4} = 30 \) times in each position. This consistency is key to further calculations, especially when we calculate the number contributions based on position.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to come to a conclusion based on given premises or known facts. In this problem, the reasoning involves ensuring every step is logically connected to yield the final answer, a sum of all possible numbers. To do this:

• Understand that creating a sum from permutations involves knowing how each number repeats across positions.
• Recognize the influence of digit placement—thousands, hundreds, tens, and units—on the final value of each number.
• Ensure calculations leverage the symmetrical contribution of each digit in every position, simplifying the total sum computation.

By applying such reasoning, it becomes clear that if you correctly calculate the contributions per position, you will correctly solve the problem by summing these contributions across all digit placements.

Number Combinations

Number combinations refer to the different ways digits can be chosen or arranged. In the context of our exercise, it is important to consider how these chosen combinations affect the overall sum we are calculating. Unlike permutations, where order matters, in combination calculations, order does not matter. However, in this problem, since each digit must be unique within each number and each has specific importance in their respective positions, the permutations effectively dictate the combinations without additional separate combination calculations. Thus, in computing the sum:

• Each digit equally contributes to the sum for every number because of the equal frequency in each position.
• The combined choice of four from five with unique permutations ensures a symmetrical sum generation pattern.

All these factors culminate to ensure the correctness in the overall sum calculation of four-digit numbers using these digits without repetition.