Question

Permutations Evaluate the following permutations. (HINT: Your scientific calculator may have a function that allows you to calculate permutations and combinations quite easily.) a. \(P_{3}^{5}\) b. \(P_{9}^{10}\) c. \(P_{6}^{6}\) d. \(P_{1}^{20}\)

Short answer

b) What is the value of \(P_{9}^{10}\)? c) What is the value of \(P_{6}^{6}\)? d) What is the value of \(P_{1}^{20}\)? a) The value of \(P_{3}^{5}\) is 60. b) The value of \(P_{9}^{10}\) is undefined, as it's not possible to arrange 10 objects in 9 positions. c) The value of \(P_{6}^{6}\) is 720. d) The value of \(P_{1}^{20}\) is 20.

Step-by-step solution

Evaluate \(P_{3}^{5}\)

Use the permutation formula to calculate \(P_{3}^{5}\): \(P_{3}^{5} = \frac{5!}{(5-3)!}\) \(P_{3}^{5} = \frac{5!}{2!}\) Now calculate the factorial of 5 and 2: \(P_{3}^{5} = \frac{120}{2}\) Finally, divide 120 by 2: \(P_{3}^{5} = 60\)

Evaluate \(P_{9}^{10}\)

It is not possible to arrange 10 objects in 9 positions, so \(P_{9}^{10}\) is undefined.

Evaluate \(P_{6}^{6}\)

Use the permutation formula to calculate \(P_{6}^{6}\): \(P_{6}^{6} = \frac{6!}{(6-6)!}\) \(P_{6}^{6} = \frac{6!}{0!}\) By convention, 0! is defined as 1, so: \(P_{6}^{6} = \frac{6!}{1}\) Now calculate the factorial of 6: \(P_{6}^{6} = \frac{720}{1}\) Finally, divide 720 by 1: \(P_{6}^{6} = 720\)

Evaluate \(P_{1}^{20}\)

Use the permutation formula to calculate \(P_{1}^{20}\): \(P_{1}^{20} = \frac{20!}{(20-1)!}\) \(P_{1}^{20} = \frac{20!}{19!}\) Now calculate the factorial of 20 and 19: \(P_{1}^{20} = \frac{2,432,902,008,176,640,000}{121,645,100,408,832,000}\) Finally, divide the values: \(P_{1}^{20} = 20\)

Key concepts

Factorial Calculation

Factorials are a key component of permutations and are denoted by the symbol "!". If you see a number followed by an exclamation point, it means you need to calculate the factorial of that number. For any positive integer \( n \), the factorial \( n! \) is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow extremely fast as \( n \) increases, making them quite large in a short span.

Another crucial point is the calculation of \( 0! \). By convention, \( 0! \) is equal to 1. This definition is useful when evaluating expressions like subsequent permutations where the number of objects chosen equals the total available, such as \( P_{6}^{6} \). Calculating these can feel tricky, but practicing different values will make it easier!

When working with permutations, understanding how to effectively calculate factorials will help streamline your work. A scientific calculator can assist with this for larger numbers to simplify the process.

Permutation Formula

The permutation formula is used to calculate the number of ways to arrange a subset of items from a larger set. The formula is expressed as:

• \( P_{r}^{n} = \frac{n!}{(n-r)!} \)

Where \( n \) is the total number of items to choose from and \( r \) is the number of items to arrange. This formula computes the number of different ordered groupings you can create, crucial for situations where the order matters, such as seating arrangements or task schedules.

Let’s take \( P_{3}^{5} \) as an example. Here, \( n = 5 \) and \( r = 3 \). By substituting into the formula, we calculate:

• \( P_{3}^{5} = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60 \)

This means there are 60 different ways to arrange 3 objects from a group of 5. Permutations can be very practical when considering combinations of components for various scenarios.

Undefined Permutation Cases

In permutations, each position needs to be filled with an item from the set. If you try to arrange more items than available, such as \( P_{9}^{10} \), it becomes undefined because it’s impossible to organize more positions than items present.

In the example \( P_{9}^{10} \), you would be attempting to organize 10 items into 9 positions. Since the number of positions exceeds the number of items, there are no valid arrangements possible. This leads to an undefined case which doesn’t provide any meaningful result.

Understanding undefined permutation cases is crucial to troubleshooting. It helps avoid errors when determining how these calculations apply in real-world or theoretical scenarios. Remember, the rule of thumb is that \( r \) cannot be greater than \( n \). If it is, reevaluate your approach.