Question

How many 6 -digit numbers can be formed using \(\\{1,2,3, \ldots, 9\\}\) with no repetitions such that 1 and 2 do not occur in consecutive positions?

Short answer

47040 numbers can be formed.

Step-by-step solution

Calculate Total Permutations

First, calculate the total number of 6-digit numbers that can be formed using the digits 1 through 9 without repetition. This is done by selecting 6 digits from 9 digits and arranging them. The number of permutations is given by: \(9 \times 8 \times 7 \times 6 \times 5 \times 4 = 60480\.\)

Calculate Permutations with 1 and 2 Together

Consider 1 and 2 as a single 'block' or 'unit' that must stay together. Treat this block as a single digit first, which gives us 5 blocks (one being the '12' block) to arrange. The number of permutations of these 5 blocks is \(8 \times 7 \times 6 \times 5 \times 4\) because there are now 8 total elements (including the separate units in the '12' block). So, the different permutations are \(8! \times 5! = 8 \times 7 \times 6 \times 5 \times 4 \). Additionally, we can arrange 1 and 2 within their block in 2 different ways, either as 12 or 21. Hence, the total permutations where 1 and 2 are together is \(2 \times (8 \times 7 \times 6 \times 5 \times 4) = 2 \times 6720 = 13440\.\)

Calculate Permutations with 1 and 2 Not Together

To find the number of permutations where 1 and 2 do not appear consecutively, subtract the number of permutations where they are together from the total number of permutations. Use the calculations from Step 1 and Step 2: \(60480 - 13440 = 47040\.\)

Key concepts

Discrete Mathematics

Discrete Mathematics is a branch of mathematics dealing with structures that are fundamentally discrete rather than continuous. This means discrete math focuses on things that can only take distinct, separated values. It is essential for computer science, as it includes topics like logic, number theory, and graph theory.

In our exercise, we're dealing with permutations, which is a discrete problem. Each solution results in a specific order of numbers, which is why we don't deal with fractions or decimals in our calculations. The calculations performed are fundamentally rooted in the discrete nature of the integers we are working with, specifically when choosing and arranging distinct digits.

Combinatorics

Combinatorics is the field of mathematics concerning the counting, arrangement, and combination of objects. In our problem, we use combinatorics to find different ways to arrange numbers without repetition, showing an arrangement—rather than a selection—of digits.

- **Total Arrangements:** We started by calculating how many 6-digit numbers you can make from the digits 1 through 9 without repeats. This involves selecting 6 digits and arranging them in order. - **Effect of Repetition:** The arrangement changes when conditions are applied, like avoiding consecutive placements of certain digits. Selective grouping and counting under constraints are fundamental parts of combinatorial problems.

Constraints in Permutations

Constraints in permutation problems add an extra layer of complexity. Constraints can include physical limits, such as prohibiting certain arrangements as seen in our problem, where we do not want the digits 1 and 2 to be adjacent. Handling constraints requires a keen understanding of how to simplify problems by shifting perspective, like viewing specific numbers as blocks.

In this scenario, considering 1 and 2 as a single unit helped us explore solutions systematically. By treating these numbers like a single block initially, we made calculations easier for fully understanding permutations with a constraint. This method simplifies the permutations task by reducing the problem into manageable parts.

Counting Methods

Counting methods are integral to solve permutation problems effectively. For our given problem, we utilized factorials and subtraction to manage constraints and derive the correct total number of permutations.

- **Factorials:** Used for determining the number of ways to arrange objects. For example, the factorial of 6, represented as 6!, is used for the total permutations possible without constraints. - **Subtraction Method:** This is seen where we start by calculating all possible scenarios and then subtracting the unwanted outcomes (where 1 and 2 are together) from the total. - **Block Method:** When constraints like '1 and 2 not being consecutive' are applied, recognizing a block simplifies the counting as it reduces the arrangement problem to fewer elements.

Using these methods in tandem allows us to navigate through complexities of permutation problems effectively and arrive at accurate solutions.