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Chapter 8: Problem 22

Evaluate each expression. $$ \frac{_{20} P_{2}}{2 !}-_{20} C_{2} $$

Short Answer

Expert verified
The solution of the given expression is 0.

Step by step solution

01

Understanding Permutation and Combination

A permutation, denoted here by _{20}P_{2}, refers to the number of ways of selecting and arranging 2 items out of 20. The formula for a permutation is _{n}P_{r} = \frac{n!}{(n-r)!}. A combination, denoted here by _{20}C_{2}, refers to the number of ways of selecting 2 items out of 20 without considering the order. The formula for a combination is _{n}C_{r} = \frac{n!}{r!(n-r)!}. So, first calculate _{20}P_{2} and _{20}C_{2}.
02

Calculate the Permutation

Using the formula for permutation, we have _{20}P_{2} = \frac{20!}{(20-2)!} = \frac{20!}{18!}. Simplifying this gives us 20 * 19 = 380.
03

Calculate the Combination

Using the formula for combination, we \(_{20}C_{2} = \frac{20!}{2!(20-2)!} = \frac{20!}{2! *18 !}\). Simplifying this gives us \(\frac{20 * 19}{2 * 1} = 190\).
04

Substitute the values into the given expression

Now substitute _{20}P_{2} and _{20}C_{2} into the original expression. This gives us \(\frac{380}{2!}-190 = \frac{380}{2*1}-190 = 190 - 190= 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

permutation formula
Permutations deal with the scenes of selecting and arranging objects in a specific order. The general permutation formula is expressed as
\[ _{n}P_{r} = \frac{n!}{(n-r)!} \]
where \( n! \) (n factorial) represents the product of all positive integers up to n, and \( (n-r)! \) denotes the factorial of the difference between the total number, n, and the number of items, r, we want to arrange. In a real-world context, understanding this is like figuring out how many different ways you could sit 3 friends in 5 chairs, considering the order important. It is vital to interpret each part of the formula clearly to master permutations, making it simpler to tackle a variety of problems.
combination formula
Unlike permutations, combinations focus on the selection of objects without considering the order in which they're arranged. The combination formula is written as
\[ _{n}C_{r} = \frac{n!}{r!(n-r)!} \]
where \( r! \) is the factorial of the items selected. This calculation finds practical uses in scenarios where outcomes are order-independent, like picking two fruits from a basket of five, regardless of which fruit is chosen first. A solid understanding of combinations helps to confidently approach problems involving groups, teams, or sets where the arrangement is not a factor.
factorial notation
Factorial notation is a mathematical shorthand to describe the product of all positive integers up to a certain number. For any non-negative integer n, the factorial is denoted by \( n! \) and is defined as
\[ n! = n \times (n-1) \times (n-2) \times ... \times 3 \times 2 \times 1 \]
For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). There's a special case where \(0! = 1\), which is a defined convention to simplify expressions involving factorials. Familiarizing yourself with factorials is crucial for manipulating permutation and combination expressions effectively.
evaluate expressions in algebra
Evaluating expressions in algebra involves substituting numbers for variables and performing the operations according to the order of operations 鈥� parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). It's like following a recipe step by step to ensure the final result is correct. When working with factorials in expressions, calculate the factorial value first before applying other operations. This process is essential for solving many algebraic problems, especially those involving permutations and combinations where factorials are a core element.

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Most popular questions from this chapter

The group should select real-world situations where the Fundamental Counting Principle can be applied. These could involve the number of possible student ID numbers on your campus, the number of possible phone numbers in your community, the number of meal options at a local restaurant, the number of ways a person in the group can select outfits for class, the number of ways a condominium can be purchased in a nearby community, and so on. Once situations have been selected, group members should determine in how many ways each part of the task can be done. Group members will need to obtain menus, find out about telephone-digit requirements in the community, count shirts, pants, shoes in closets, visit condominium sales offices, and so on. Once the group reassembles, apply the Fundamental Counting Principle to determine the number of available options in each situation. Because these numbers may be quite large, use a calculator.

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Exercises \(95-97\) will help you prepare for the material covered in the next section. The figure shows that when a die is rolled, there are six equally likely outcomes: \(1,2,3,4,5,\) or \(6 .\) Use this information to solve each exercise. (image can't copy) What fraction of the outcomes is not less than \(5 ?\)

A mathematics exam consists of 10 multiple-choice questions and 5 open-ended problems in which all work must be shown. If an examinee must answer 8 of the multiple-choice questions and 3 of the open-ended problems, in how many ways can the questions and problems be chosen?

Solve by the method of your choice. Using 15 flavors of ice cream, how many cones with three different flavors can you create if it is important to you which flavor goes on the top, middle, and bottom?
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