Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$P(7,7)$$

Step-by-Step Solution

Answer

There are 5040 permutations when arranging 7 items taken all at once.

Step 1: Understand the Permutation Notation

The expression \( P(n,r) \) represents the number of permutations of \( n \) items taken \( r \) at a time. In this case, \( P(7,7) \) means the number of ways to arrange 7 items taken all 7 at a time.

Step 2: Recall the Permutation Formula

The permutation formula is given by \( P(n,r) = \frac{n!}{(n-r)!} \). We will use this formula to calculate \( P(7,7) \).

Step 3: Substitute Values into the Formula

Substitute \( n = 7 \) and \( r = 7 \) into the permutation formula: \[ P(7,7) = \frac{7!}{(7-7)!} = \frac{7!}{0!} \]

Step 4: Calculate Factorials

Calculate \( 7! \) and \( 0! \):- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)- \( 0! = 1 \) (by definition)

Step 5: Compute the Permutation Value

Now insert the factorials into the expression: \[ P(7,7) = \frac{5040}{1} = 5040 \]

Step 6: Interpret the Result

Since \( P(7,7) = 5040 \), there are 5040 ways to arrange 7 items taking all 7 items at once.

Key Concepts

Factorials

Factorials are fundamental to understanding permutations, and they are often represented by the symbol "!". In mathematics, the factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \).
For example, the factorial of 5, denoted as \( 5! \), is calculated as follows:

• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

A special case to remember is that \(0!\) is by definition 1. This might seem odd initially, but it helps maintain consistency in mathematical expressions, especially permutations and combinations.
Factorials grow very quickly as the number increases. By understanding how factorials operate, you can solve problems involving permutations more efficiently.

Arranging Items

Arranging items is a common way to think about permutations. Permutations are different possible sequences or orders in which a set or subset of items can be arranged.
This is crucial when you want to determine how many ways you can organize a number of distinct items, like letters, numbers, or objects. To visualize why order is important, consider arranging 3 distinct letters: A, B, and C. Here are their permutations:

• ABC
• ACB
• BAC
• BCA
• CAB
• CBA

There are 6 different ways to order these 3 items, which aligns with calculating \(3!\). This demonstrates the variety in arranging items when order matters.
When each item in the set must be included in every arrangement, permutations provide a solid method to explore all possibilities systematically.

Permutation Formula

To determine the number of ways to arrange a subset of items from a larger set, we use the permutation formula \(P(n,r) = \frac{n!}{(n-r)!}\). In this formula:

• \(n\) is the total number of items.
• \(r\) is the number of items to arrange.

The formula's logic rests on the principle of multiplying decreasing choices as each item is selected. For example, if we have 7 items and need to arrange all 7, it becomes \(P(7,7)\). By applying the formula:

• Plug values into the formula: \(P(7,7) = \frac{7!}{(7-7)!} = \frac{7!}{0!}\)
• Calculate \(7! = 5040\)
• Recall \(0! = 1\)
• Thus, \(P(7,7) = 5040\)

This formula highlights the influence of factorials in permutations and underscores the importance of the sequence arrangement. So, for 7 distinct items, arranging them all is like calculating a simple factorial.