Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to evaluate each expression. $$_{15} P_{8}$$

Step-by-Step Solution

Answer

The value of \(_{15}P_{8}\) is 2,533,440.

Step 1: Understand the Permutation Notation

The permutation notation \(_{n}P_{r}\) represents the number of ways to choose and arrange \(r\) items from \(n\) items. It is calculated using the formula \(_{n}P_{r} = \frac{n!}{(n-r)!}\). For the expression \(_{15}P_{8}\), \(n = 15\) and \(r = 8\).

Step 2: Calculate the Factorial of n

Compute \(15!\) (15 factorial), which is the product of all positive integers from 1 to 15:\[15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\]

Step 3: Calculate the Factorial of (n-r)

Compute \((15-8)!\), which is \(7!\) (7 factorial), the product of all positive integers from 1 to 7:\[7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\]

Step 4: Divide the Factorials

Use the permutation formula to find the result of \(_{15}P_{8}\):\[_{15}P_{8} = \frac{15!}{7!}\]

Step 5: Simplify and Calculate

Simplify the expression by canceling out \(7!\) from \(15!\):\[_{15}P_{8} = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8\]Calculate this product using a calculator to get the final result.

Key Concepts

Factorial

A factorial, denoted by a "!" after a number, is the product of that number and every positive integer below it. Factorials are fundamental in permutations and other areas in mathematics as they help in calculating the number of possible different arrangements. For example, when you see a notation like \(15!\), it means:

• Multiply 15 by 14, by 13, repeatedly until 1.
• This translates to: \[15! = 15 \times 14 \times 13 \times \ldots \times 1\]

Using this process helps in breaking down more complex problems into simple multiplication tasks. Factorials increase very quickly, making large numbers from seemingly small starting numbers.

Permutation formula

Permutations focus on counting the number of possible arrangements of a set of objects. When the order matters, as it does in permutations, a specific formula is useful. The formula used for permutations is \(_{n}P_{r} = \frac{n!}{(n-r)!}\). This helps identify how many different sequences of \(r\) items can be formed from \(n\) total items.

• \(n!\) represents the factorial of the total number of items, providing the total possible arrangements.
• \((n-r)!\) accounts for the items not selected, removing them from the arrangements.

Thus, the formula efficiently calculates possible orders, ensuring everything left after selection is ignored.

Calculator usage

Calculating permutations manually requires multiple steps and can be time-consuming. This is where calculators become handy, especially scientific or graphing calculators which often have built-in factorial and permutation functions. Here's how you can use a calculator for permutations:

• Locate the factorial (\(!\)) key, often under a math or probability menu.
• Enter the first number \(n\), then use the permutation or factorial function to compute \(n!\).
• Repeat for \( (n - r)!\), if not built-in.
• Your calculator may directly compute \(_{n}P_{r}\) if it has this function, generally accessible through the probability features.

Calculators simplify complex multiplication into a button-press activity, reducing the risk of arithmetic errors.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. This area of mathematics includes permutations as a way of understanding how different arrangements are possible when the order is a factor. Through combinatorics:

• We explore arrangements and select groups of items from larger sets.
• Permutations allow us to find various sequences from a set group, while combinations neglect order.
• Understanding how portions of sets relate back to the full set is integral, using tools like permutations and combinations to solve problems.

By mastering the basics of combinatorics, one gains deeper insight into the practical ways of arranging and combining entities, which is a skill applicable in diverse fields.