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Chapter 10: Problem 9

A product like \(3 \cdot 2 \cdot 1\) or \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\) is called a factorial expression and is written with an exclamation point, like this: \(3 \cdot 2 \cdot 1=3\) ! and \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=5\) !. a. How can you calculate 8 !? b. How can you use factorial notation to calculate the number of permutations of 10 objects chosen 10 at a time? \([-\square\) See Calculator Note \(10 \mathrm{~F}\) to leam how to compute \(n\) with your calculator. 4 ] c. Write an expression in factorial notation that can be used to calculate \({ }_{n} P_{n}\).

Short Answer

Expert verified
a. 8! = 40320. b. P(10, 10) = 10! = 3628800. c. \( {}_{n}P_{n} = n! \).

Step by step solution

01

Understanding Factorials

The factorial of a number, denoted as \( n! \), is the product of all positive integers from 1 to \( n \). For example, \( 3! = 3 \times 2 \times 1 = 6 \). This basic understanding will help solve the problems given in the exercise.
02

Calculating 8!

To calculate \( 8! \), multiply all integers from 1 to 8. This can be written as follows:\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320. \]
03

Understanding Permutations

Permutations are arrangements of objects in a specific order. The number of permutations of \( n \) objects taken \( r \) at a time is given by \( P(n, r) = \frac{n!}{(n-r)!}. \) This formula represents the product of \( n \) descending numbers divided by the product of numbers after \( (n-r) \).
04

Calculating Permutations for 10 Objects

For 10 objects chosen 10 at a time, the permutations are calculated using:\[ P(10, 10) = \frac{10!}{(10-10)!} = \frac{10!}{0!}. \]Since \( 0! = 1 \), this simplifies to \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800. \)
05

Expression in Factorial Notation

To express permutations of \( n \) objects taken \( n \) at a time, the formula is: \[ {}_{n}P_{n} = \frac{n!}{(n-n)!} = \frac{n!}{0!}. \] Simplifying the expression, we have \( {}_{n}P_{n} = n! \) because \( 0! = 1 \).

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Key Concepts

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

Permutations
Permutations are a fundamental concept in mathematics, particularly in combinatorics, which deals with counting and arrangement patterns. If you've ever stood in front of your wardrobe wondering how many different outfits you can make, you've already engaged with permutations! In more formal terms, permutations refer to different ways we can arrange a set of objects in a particular order.
  • Specific Order: The order in which the objects are arranged is crucial 鈥� swapping two objects creates a different permutation.
  • Formula Insight: The formula for calculating permutations of n objects taken r at a time is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \]
  • Application: When you need to arrange a group of 10 people in a line, you'd use \( P(10, 10) \), meaning you are arranging all 10 objects fully.
Understanding permutations is crucial in solving many practical problems, from scheduling to cryptography.
Factorial Notation
In mathematics, factorial notation is a shorthand method of writing a product of descending natural numbers. It's denoted by an exclamation mark (!) following an integer. This might seem surprising at first, but once you get used to it, it simplifies calculations significantly.
  • Notation: The factorial of a number \( n \) is written as \( n! \).
  • Calculation: To find \( n! \), multiply all whole numbers from \( n \) down to 1. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
  • Zero Factorial: By definition, \( 0! = 1 \) which might seem puzzling at first. However, it is agreed upon to maintain the consistency of certain mathematical formulas, especially in combinatorics.
Using factorial notation helps in simplifying complex expressions and is integral in permutations and combinations.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's like the toolkit providing the means to formulate equations and solve problems involving unknown numbers.
  • Application in Permutations: Algebraic manipulation is key in understanding permutations. By using algebraic expressions such as \( \frac{n!}{(n-r)!} \), we can calculate permutations efficiently.
  • Solving Equations: Algebra allows us to express general terms and solve for unknowns, making it indispensable in higher-level mathematics.
  • Variables and Constants: Learning to work with these elements helps in simplifying expressions and understanding the relationships between numbers.
Algebra forms the base for almost all mathematical concepts, including the interpretation and solution of complex factorial and permutation problems.
Mathematical Expressions
Mathematical expressions are combinations of numbers, symbols, and operators (such as +, 鈭�, 脳, 梅) that represent a value or describe a relationship. These expressions are the language of mathematics and serve as a bridge between rigorous mathematical theories and practical applications.
  • Elements: An expression can include constants, variables, functions, operators, and also factorials like \( n! \).
  • Usage in Exercises: Writing expressions in factorial notation simplifies the representation of larger problems, such as creating a compact formula to calculate \( {}_{n}P_{n} \).
  • Simplification: By learning to manipulate and simplify expressions, you can solve problems faster and more efficiently.
Familiarity with mathematical expressions allows students to navigate between numerical answers and the abstract concepts they represent.

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Most popular questions from this chapter

Draw and label a segment like this one. Plot and label points on your segment to represent the probability for each situation. a. You will eat breakfast tomorrow morning. b. It will rain or snow sometime during the next month in your hometown. c. You will be absent from school fewer than five days this school year. d. You will get an A on your next mathematics test. e. The next person to walk in the door will be under 30 years old. f. Next Monday every teacher at your school will give 100 free points to each student. g. Earth will rotate once on its axis in the next 24 hours.

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APPLICATION The star hitter on the baseball team at City Community College had a batting average of 375 before the start of a three-game series. (Note: Batting average is calculated by dividing hits by times at bat; sacrifice bunts and walks do not count as times at bat.) During the three games, he came to the plate to bat eleven times. In these eleven plate appearances, he walked twice and had one sacrifice bunt. He either got a hit or struck out in his other plate appearances. If his batting average was the same at the end of the three-game series as at the beginning how many hits did he get?

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