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Chapter 11: Problem 24

Evaluate each factorial expression. \(6 !-3 !\)

Short Answer

Expert verified
The result is 714.

Step by step solution

01

Calculate the Factorial of 6

First calculate the factorial for the number 6. \(6 !\) translates to \(6 脳 5 脳 4 脳 3 脳 2 脳 1 = 720\)
02

Calculate the Factorial of 3

Next, calculate the factorial for the number 3. To do this, we take \(3 !\) that translates to \(3 脳 2 脳 1 = 6\)
03

Subtract the results

Finally, subtract the result of the factorial of 3 from the result of the factorial of 6: \(720 - 6 = 714\).

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Key Concepts

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

Subtraction
Subtraction is one of the four basic arithmetic operations, usually denoting the operation of removing objects from a collection. In our specific exercise, subtraction becomes the final step after calculating factorials for two numbers. Here, subtraction is used to find the difference between two factorial values:
  • Calculate each number's factorial
  • Subtract the smaller factorial value from the larger factorial value
The operation can seem straightforward, but don鈥檛 underestimate it. Knowing when and how to apply subtraction is crucial, especially when working on math problems involving more complex operations like factorials. Remember to always align numbers correctly and double-check your subtraction to avoid any mishaps.
Factorial Expression
A factorial expression in mathematics involves multiplying a series of descending natural numbers. The symbol for factorial is an exclamation mark (!). For example, calculating a factorial involves determining \( n! = n imes (n-1) imes (n-2) imes ext{...} imes 1\).An important aspect of factorials is understanding what it signifies: the total number of ways to arrange "n" items in an order. In our exercise, we calculated:
  • For 6: \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)
  • For 3: \( 3! = 3 \times 2 \times 1 = 6\)
Knowing how to handle such calculations accurately is a keystone in solving larger mathematical expressions and problems.
Math Problem Solving
Math problem solving is a systematic process used to find solutions to complex calculations and equations. The main steps involve understanding the problem, planning how to solve it, executing the plan, and reviewing the solution. In solving factorial expressions:
  • Understand what a factorial is and how it applies to the numbers given.
  • Execute the multiplication steps accurately.
  • Apply subtraction correctly for the final solution.
  • Ensure each step checks out by revisiting and reviewing your calculations.
Structured problem solving not only improves accuracy but also builds a strong foundation for tackling even more complex math issues.
Arithmetic Operations
Arithmetic operations form the cornerstone of mathematics. They include addition, subtraction, multiplication, and division. Each fulfills a specific role and is used in sequence to solve mathematical problems. In factorial evaluations, multiplication is used to unfold the factorial expression. Afterward, subtraction helps to derive the final result by comparing and "taking away" one factorial from another.
  • Multiplication: Essential for calculating the continuous product in factorials.
  • Subtraction: Used to find differences between calculated values.
Understanding these operations deeply helps simplify and solve problems efficiently. Always review your work to ensure each operation has been correctly applied.

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Most popular questions from this chapter

One card is randomly selected from a deck of cards. Find the odds in favor of drawing a red card.

In Exercises 15-20, you draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a picture card the first time and a heart the second time.

Involve computing expected values in games of chance. Another option in a roulette game (see Example 6 on page 760 ) is to bet \(\$ 1\) on red. (There are 18 red compartments, 18 black compartments, and 2 compartments that are neither red nor black.) If the ball lands on red, you get to keep the \(\$ 1\) that you paid to play the game and you are awarded \(\$ 1\). If the ball lands elsewhere, you are awarded nothing and the \(\$ 1\) that you bet is collected. Find the expected value for playing roulette if you bet \(\$ 1\) on red. Describe what this number means.

Explain how to find and probabilities with dependent events. Give an example.

A popular state lottery is the \(5 / 35\) lottery, played in Arizona, Connecticut, Illinois, Iowa, Kentucky, Maine, Massachusetts, New Hampshire, South Dakota, and Vermont. In Arizona's version of the game, prizes are set: First prize is \(\$ 50,000\), second prize is \(\$ 500\), and third prize is \(\$ 5\). To win first prize, you must select all five of the winning numbers, numbered from 1 to 35 . Second prize is awarded to players who select any four of the five winning numbers, and third prize is awarded to players who select any three of the winning numbers. The cost to purchase a lottery ticket is \(\$ 1\). Find the expected value of Arizona's "Fantasy Five" game, and describe what this means in terms of buying a lottery ticket over the long run.
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