Problem 39 - Algebra for College Students

Question

Evaluate each expression. \(P(8,0)\)

Step-by-Step Solution

Answer

\(P(8, 0) = 1\)

Step 1: Understand the Permutation Notation

The permutation notation \(P(n, r)\) represents the number of ways to arrange \(r\) objects from a set of \(n\) objects in a specific order.

Step 2: Identify the Given Values

In the expression \(P(8, 0)\), \(n = 8\) and \(r = 0\). We need to find the number of ways to arrange 0 objects from a set of 8 objects.

Step 3: Apply the Permutation Formula

The formula for permutations is \(P(n, r) = \frac{n!}{(n-r)!}\). Substituting the given values, we have: \[P(8, 0) = \frac{8!}{(8-0)!} = \frac{8!}{8!}\].

Step 4: Simplify the Expression

Since \(8!\) divided by \(8!\) equals 1, we have \[P(8,0) = 1\].

Key Concepts

Permutation Formula

Permutations are a fundamental concept in combinatorics, which is the branch of mathematics dealing with counting and arrangement. To find how many ways we can arrange a certain number of objects from a larger set, we use the permutation formula.
The general permutation formula is as follows:
\[\begin{equation} P(n, r) = \frac{n!}{(n-r)!} \ \text{where} \ n = \text{total number of objects} \ r = \text{number of objects to arrange} \ ! = \text{factorial} \ P(n, r) = \text{the number of permutations} \ \text{of } r \text{ objects from } n \text{ objects} \ \text{in specific order} \ \text{Note}: \ 0! = \text{1}\text{ as a special case} \ \text{Using this,}\ P(8, 0) = \frac{8!}{(8-0)!} = \frac{8!}{8!} = 1 \ \text{Only one way to arrange 0 objects from a set of 8}\text.{ \end{equation}\]} \ \text{The formula helps in various problems where the sequence of arrangement is crucial.}

Factorial

Factorial is another key concept related to permutations. The factorial of a number, denoted by a '!', is the product of all positive integers up to that number.
Example: \[\begin{equation} n! = n \times (n-1) \times (n-2) \times ... \times 3 \times 2 \times 1 \ \text{For instance, 4! = 4 × 3 × 2 × 1 = 24} \ \text{Factorial grows very fast} \ \text{5! = 120, 6! = 720, etc.} \ \text{Special Case:} \ 0! = 1 \text{(by definition)} \end{equation}\]
Factorials are used in the permutation formula to determine different ways of arranging objects. The factorial's properties make it efficient for calculations in combinatorial problems.

Arrangement of Objects

The arrangement of objects in permutations refers to how objects can be ordered or sequenced.
When considering permutations, both the set size and the arrangement matters. If the order did not matter, then we would be dealing with combinations instead of permutations.
Example:
Arranging 3 objects A, B, and C:\[\begin{equation} \text{ABC, ACB, BAC, BCA, CAB, CBA} \ \text{6 different ways this shows how sequence matters in permutations} \end{equation}\]
In the case of the exercise, the arrangement of 0 objects from 8 means there's only 1 ('empty') way to arrange them, highlighting the unique nature of permutations when dealing with a count of zero.
Understanding these arrangements helps solve problems involving permutation with clarity and accuracy.