Question

Evaluate the expression. \({ }_{12} P_0\)

Short answer

The value of \({ }_{12} P_0\) is 1.

Step-by-step solution

Identify the permutation

Here we have a permutation \({ }_{12} P_0\), where 12 represents the total number of items and 0 represents the number of items we want to arrange in a specific order.

Apply the permutation rule

Now, we need to apply the rule for permutations. Permutations are calculated by the formula \(Npr = \frac{n!}{(n-r)!}\) where 'n' is the total number of items, 'r' is the number of items to choose, and '!' denotes a factorial, which means multiplying all positive integers from 1 up to that number. However, any number of elements can be arranged in 1 way only, so whenever r=0, regardless of the n value, \(nPr = 1\).

Find the value

So, the value of \({ }_{12} P_0\) is 1, regardless of what the total number of items is (in this case 12).

Key concepts

Permutation Formula

When dealing with arrangements or ordering different items, we encounter a powerful concept known as permutations. A permutation is a particular arrangement of a set of objects in a specific order. Often the focus is on the number of different ways to arrange a subset of these objects. To calculate this mathematically, we use the permutation formula.

The general permutation formula is represented as \(nPr = \frac{n!}{(n-r)!}\), where

• \(n\)
is the total number of items,
• \(r\)
is the number of items to arrange,
• and the exclamation point (\(!\)) stands for factorial notation, which we'll cover in more detail later.

Using this formula, you can calculate the number of possible arrangements (\(nPr\)) for any given values of \(n\) and \(r\). For instance, if we want to know how many ways there are to arrange 5 books on a shelf taking 3 at a time, we'd compute \(5P3 = \frac{5!}{(5-3)!}\), which simplifies to \(5 \times 4 \times 3\).

An essential point to remember is that whenever \(r = 0\), the number of ways to arrange zero items is always just one way, regardless of the value of \(n\). This explains why in our original exercise, \({}_{12}P_0 = 1\).

Factorial Notation

Factorial notation plays a critical role in permutations and combinatorics. The notation itself is simple: the factorial of a non-negative integer \(n\) is the product of all positive integers less than or equal to \(n\). It's represented by the exclamation mark (\(!\)).

Here are a few examples of factorial notation:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\),
• \(3! = 3 \times 2 \times 1 = 6\),
• \(1! = 1\)

and importantly,

\(0! = 1\)

, by definition.

This last point is particularly noteworthy because it comes into play when evaluating expressions where the number of items to choose is zero, such as in our exercise \({}_{12}P_0\). The factorial of 0 being equal to 1 is not an intuitive concept, but it's a defined property that makes calculations in permutations and combinatorics work out neatly. The factorial function grows rapidly with larger values of \(n\), which reflects the quickly increasing number of permutations as more items are available to arrange.

Combinatorics

Combinatorics is the branch of mathematics focusing on counting, arranging, and the structure of configurations. It is fundamentally about finding the number of different combinations and permutations of a set of objects. While permutations are concerned with the arrangement of objects where the order is important, combinatorics also deals with combinations where the order does not matter.

In combinatorial problems, you might encounter questions like, 'How many different groups of students can you form from a class of 30?' This does not involve the order in which the students are grouped but merely the selection itself. Combinatorics uses principles and formulas like permutations (\(nPr\)) and combinations (\(nCr = \frac{n!}{r!(n-r)!}\)) to solve such problems.

Understanding the foundational concepts of permutations and factorial notation is critical in combinatorics as they often provide the tools necessary to approach and solve complex problems involving the counting of arrangements and selections. With combinatorics, you can explore more than just linear arrangements; it also encompasses structures like graphs, tessellations, and more, allowing for broad analysis of combinatorial structures in mathematics.