Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, compute the value of the expression. $$ P(11,5) $$

Step-by-Step Solution

Answer

The value of the expression is 55440.

Step 1: Understand the Permutation Notation

The notation \( P(n, r) \) denotes a permutation, which represents the number of ways to choose \( r \) objects from \( n \) objects in a specific order. It is calculated using the formula \( P(n, r) = \frac{n!}{(n-r)!} \).

Step 2: Identify n and r

Identify the values of \( n \) and \( r \) in \( P(11,5) \). Here, \( n = 11 \) and \( r = 5 \).

Step 3: Calculate n - r

Subtract \( r \) from \( n \): \( 11 - 5 = 6 \).

Step 4: Calculate n!

Calculate \( 11! \), which is the factorial of 11: \( 11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39916800 \).

Step 5: Calculate (n - r)!

Calculate \( 6! \), which is the factorial of 6: \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \).

Step 6: Use the Permutation Formula

Substitute the values into the permutation formula to calculate \( P(11,5) \): \( P(11,5) = \frac{11!}{6!} = \frac{39916800}{720} \).

Step 7: Simplify the Expression

Simplify \( \frac{39916800}{720} \) to find the value of the permutation: \( P(11,5) = 55440 \).

Key Concepts

Factorial calculations

Factorials are a fundamental concept in permutations and many other areas of mathematics. Understanding them is essential when working with permutations.

The factorial of a number, represented by an exclamation mark (!), is the product of all positive integers up to that number. For instance, to calculate the factorial of 11, written as \(11!\), you would multiply all whole numbers from 1 to 11 together:

\[ 11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

This results in 39,916,800.

Factorials grow very quickly as the number increases. Even relatively small numbers result in quite large factorials, which is crucial when performing calculations in permutations.

When calculating permutations, you often need to find the factorials of both \(n\) and \((n-r)\), where \(n\) is the total number of items and \(r\) is the number of items taken at a time.

Permutation formula

The permutation formula is a key element in determining how many ways you can order a set of items when the order matters. This is essential in problems involving arrangements or sequences.

The formula to find permutations is represented by \( P(n, r) \), which stands for the number of ways to arrange \(r\) items out of \(n\) possible items. The individual formula is:

\[ P(n, r) = \frac{n!}{(n-r)!} \]

This equation involves two factorials:

• \(n!\) - the factorial of the total number of objects.
• \((n-r)!\) - the factorial of the difference between the total objects and those selected.

By dividing \(n!\) by \((n-r)!\), you account for all possible arrangements of the \(r\) selected items, considering the total number \(n\) items.

This division is crucial as it eliminates the arrangements of the remaining items, focusing solely on the chosen \(r\) items.

Order in permutations

Order is a significant concept in permutations, setting it apart from combinations where order does not matter.

In permutations, not only are the items selected important, but also the sequence in which they are arranged.

Consider a simple example: imagine arranging letters A, B, and C.

• ABC
• ACB
• BAC

All of these represent different permutations because only the order differs.

Changing the positions changes the permutation. Suppose you have 11 people, and you need to award three prizes.

The sequence in which these prizes are given is significant because each arrangement results in a unique outcome.

Therefore, permutation calculations consider every individual order as a distinct possibility.

This aspect of permutations is essential when dealing with real-world scenarios, like scheduling or organizing events, where the sequence impacts the final result.