Question

Evaluate the permutations. $$ P_{6}^{6} $$

Short answer

Based on the given step-by-step solution, the final answer for evaluating the permutation \(P_{6}^{6}\) is 720. This means there are 720 ways to arrange 6 objects in a set containing 6 objects.

Step-by-step solution

Calculate the Factorials

First, we have to calculate the factorials for \(6\) and \(0\). For \(6!\), we will calculate: $$ 6! = 6\times 5\times 4\times 3\times 2\times 1 = 720 $$ For \(0!\), by definition: $$ 0! = 1 $$ Now we can plug these values back into our permutation formula: $$ P_{6}^{6} = \frac{720}{1} $$

Simplify the Result

Since the denominator is \(1\), we can simplify the expression to: $$ P_{6}^{6} = 720 $$ So, there are 720 ways to arrange 6 objects in a set containing 6 objects.

Key concepts

Understanding Factorials

The concept of factorials is foundational in exploring permutations and combinations, which are essential in probability and statistics. A factorial of a non-negative integer, denoted as \( n! \), is the product of all positive integers from \( n \) down to 1. In simple terms, it represents the number of ways to arrange \( n \) objects in a particular order.

For example, the factorial of 6, or \( 6! \), is calculated as \( 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \). This signifies there are 720 different ways to arrange 6 distinct objects. An interesting and special case is \( 0! \), which is always defined to be equal to 1. This isn't intuitive at first, but it makes sense within various mathematical formulas, including those of permutations and combinations, to ensure they work properly for all numbers.

Arrangement of Objects

When we discuss the arrangement of objects, we are often referring to the number of permutations of those objects. Permutations relate to the arrangement of a set of objects in a particular order. The permutation formula given as \( P_n^r = \frac{n!}{(n - r)!} \) permits us to calculate the number of ways we can arrange a subset of \( r \) objects from a larger set of \( n \) objects. It's important to note that the order of arrangement is critical in permutations.

Using our exercise as an example, \( P_6^6 \) evaluates the number of ways to arrange all 6 objects from a set of 6 objects. With permutations, since the order matters, '123456' is different from '654321', even though they contain the same objects. Hence, by calculating the total permutations, we are effectively finding all the possible orders of a given set.

The Role of Probability and Statistics

In the broader view, permutations play a crucial role in probability and statistics. Understanding permutations enables us to calculate the likelihood of different outcomes occurring, assuming all outcomes are equally likely. Probability is concerned with predicting the chance of a future event, while statistics often involves analyzing past data for forecasting.

In probability, we might use permutations to answer questions like: 'What is the probability of arranging a set of books on a shelf in a specific order?' Statistics might leverage permutations in areas such as experimental design, where understanding all possible orders of a treatment can be crucial. Thus, permutations inform both theorizing outcomes in probability and interpreting data in statistics.