Problem 26 - Algebra and Trigonometry

Question

List all the permutations of 6 objects \(1,2,3,4,5,\) and 6 choosing 3 at a time without repetition. What is \(P(6,3) ?\)

Step-by-Step Solution

Answer

There are 120 permutations: \( P(6,3) = 120 \).

Step 1 - Understand the Problem

The problem asks to list all possible permutations of selecting 3 objects out of 6 without repetition. Additionally, calculate the number of such permutations, denoted as \(P(6,3)\).

Step 2 - Define Permutation Formula

The formula for permutations of selecting \(r\) objects out of \(n\) objects is given by: \[ P(n,r) = \frac{n!}{(n-r)!} \] where \(n!\) denotes the factorial of \(n\).

Step 3 - Apply the Formula

For the given problem, \( n = 6 \) and \( r = 3 \). Applying the values in the formula, we get: \[ P(6,3) = \frac{6!}{(6-3)!} = \frac{6!}{3!} \]

Step 4 - Simplify the Expression

Calculate the factorials: \( 6! = 720 \) and \( 3! = 6 \). Thus, \[ P(6,3) = \frac{720}{6} = 120 \]

Step 5 - List All Permutations

The list of all possible permutations of 3 objects out of 6 can be written exhaustively. Some of the permutations include, for example: \( (1,2,3), (1,2,4), (1,2,5), (1,2,6), (1,3,2), (1,3,4), \) and so on. An exhaustive listing would include 120 such permutations.

Key Concepts

Factorials

A factorial, denoted by the symbol !, represents the product of all positive integers up to a given number. For example, 5! equals 5 × 4 × 3 × 2 × 1, which is 120. The general formula for calculating a factorial is: \[ n! = n × (n-1) × (n-2) × ... × 1 \] Factorials grow very quickly as N increases, making them a significant concept in permutations and combinations. Factorials are fundamental in counting problems across various fields of mathematics and science.

Permutation Formula

The permutation formula is used when we are interested in the arrangement of r objects from a set of n objects, without repetition. The formula is given by: \[ P(n,r) = \frac{n!}{(n-r)!} \] Here, n! represents the factorial of n, and (n-r)! is the factorial of the difference between n and r. This formula calculates the number of possible ways to arrange or order r objects out of n. Permutations pay attention to order, which is a major difference from combinations. Using the permutation formula is practical for tasks such as ranking items, arranging books, and scheduling.

Combinatorial Mathematics

Combinatorial mathematics, often simply called combinatorics, is a branch of mathematics focused on counting, arranging, and finding patterns. It includes the study of:

• Permutations
• Combinations
• Graph theory
• Configurations of objects

Combinatorics is largely used in fields like computer science, statistics, and even genetics. Understanding permutations (order matters) and combinations (order doesn’t matter) is vital for solving many practical and theoretical problems in combinatorics.

Permutations Without Repetition

Permutations without repetition mean that once an object is chosen, it cannot be chosen again. For example, choosing 3 out of 6 objects {1,2,3,4,5,6} without repetition involves listing every possible arrangement without repeating any object. This is calculated using P(n,r).
Let's apply it:
For P(6,3), we use the formula: \[ P(6,3) = \frac{6!}{(6-3)!} = \frac{720}{6} = 120 \] Thus, there are 120 different ways to arrange 3 out of 6 objects without repetition. Counting permutations becomes simpler using the rule and avoids exhaustive listing.