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Chapter 11: Problem 32

Evaluate each factorial expression. \(\frac{17 !}{(17-3) !}\)

Short Answer

Expert verified
The evaluated expression is 4080

Step by step solution

01

Understanding Factorials

A factorial is the product of an integer and all the integers below it. The factorial function can be symbolized by an exclamation point (!). For example, 5! is equivalent to 5 * 4 * 3 * 2 * 1.
02

Applying the Factorial Definition

The factorial functional property where the factorial of n is n times the factorial of (n-1) can be used to simplify this expression. Using this property, the expression \(17!/(17-3)!\) can also be written as \(17*16*15*14!\) divided by \(14!\).
03

Simplification

Now, \(14!\) in the numerator and denominator cancels out each other, leaving us with \(17*16*15\). Amalgamating these gives us an end result of 4080.

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Key Concepts

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

Factorial Definition
In mathematics, the concept of a factorial is fundamental. It's defined as the product of an integer and all the positive integers less than it. This operation is symbolized by an exclamation mark (!). For instance, the factorial of 5, written as 5!, can be calculated as:
  • 5 x 4 x 3 x 2 x 1 = 120
This demonstrates the concept of a decreasing multiplicative sequence. Thus, the factorial operation is a way of multiplying a series of descending natural numbers. Factorials are typically used in permutations, combinations, and other areas of discrete mathematics. When you see a number followed by an exclamation mark, remember, it's time to start multiplying down to 1!
Factorial Simplification
The factorial simplification process can dramatically ease computations, especially when dealing with factorials in division. Imagine you have an expression involving factorials in the form of a fraction, like \[\frac{n!}{k!}\]where \(n > k\). You can often cancel terms to simplify the expression. Consider our exercise: \[\frac{17!}{14!}\]Here, you apply the property that \(n!\) is \(n \times (n-1)!\) recursively. Thus:
  • \(17! = 17 \times 16 \times 15 \times 14!\)
With both numerator and denominator containing \(14!\), we can cancel out these terms, simplifying our expression to:
  • \(17 \times 16 \times 15 \)
This cancellation is akin to removing common factors in fractions, streamlining calculations down to manageable integer multiplications.
Mathematical Expressions
Mathematical expressions involving factorials can initially appear daunting, but they follow logical rules. In our example, we dealt with a factorial expression in the form:
  • \(\frac{17!}{(17-3)!}\)
The subtraction in the denominator indicates a reduced factorial sequence. Breaking it down with simple arithmetic operations transforms the problem into smaller, manageable chunks. With factorials:
  • The numerator \(17!\) contains all products from 17 downwards, while the denominator \(14!\) ends at 14.
By understanding how to decompose these expressions, you can efficiently reach simplified forms and compute results quickly. This clarity allows you to approach seemingly complex mathematical expressions with confidence, unraveling them into simpler computations.

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Most popular questions from this chapter

Involve computing expected values in games of chance. A game is played using one die. If the die is rolled and shows 1 , the player wins \(\$ 1\); if 2 , the player wins \(\$ 2\); if 3 , the player wins \(\$ 3\). If the die shows 4,5 , or 6 , the player wins nothing. If there is a charge of \(\$ 1.25\) to play the game, what is the game's expected value? What does this value mean?

We return to our box of chocolates. There are 30 chocolates in the box, all identically shaped. Five are filled with coconut, 10 with caramel, and 15 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting a coconut-filled chocolate followed by a solid chocolate.

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

In a class of 50 students, 29 are Democrats, 11 are business majors, and 5 of the business majors are Democrats. If one student is randomly selected from the class, find the probability of choosing a Democrat or a business major.

A 25 -year-old can purchase a one-year life insurance policy for \(\$ 10,000\) at a cost of \(\$ 100\). Past history indicates that the probability of a person dying at age 25 is \(0.002\). Determine the company's expected gain per policy.
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