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Chapter 8: Problem 23

Evaluate each factorial expression. $$\frac{17 !}{15 !}$$

Short Answer

Expert verified
The value of \( \frac{17!}{15!} \) is 272.

Step by step solution

01

Understand Factorials

Factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 17! = 17 脳 16 脳 15 脳 14 脳 ... 脳 2 脳 1.
02

Break Down Terms

We break down the factorial terms one by one without actually calculating the values. So, \( \frac{17!}{15!} \) becomes \( \frac{17 脳 16 脳 15 脳 14 脳 ... 脳 2 脳 1 }{15 脳 14 脳 ... 脳 2 脳 1} \).
03

Simplify Expression

As seen above, both numerator and denominator have common factors. They cancel each other. What's left in the expression after cancelling out the common terms would be \( \frac{17 脳 16}{1} \).
04

Calculate Result

After simplification we multiply the remaining non-cancelled terms in the numerator to get our result. That would be 17 * 16 = 272.

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Most popular questions from this chapter

Exercises \(67-72\) are based on the following jokes about books: \(\cdot\) "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." - Groucho Marx \(\cdot\) "I recently bought a book of free verse. For \(\$ 12\)." \- George Carlin \(\cdot\) "If a word in the dictionary was misspelled, how would we know?" - Steven Wright \(\cdot\) "Encyclopedia is a Latin term. It means 'to paraphrase a term paper." - Greg Ray \(\cdot\) "A bookstore is one of the only pieces of evidence we have that people are still thinking." - Jerry Seinfeld \(\cdot\) "I honestly believe there is absolutely nothing like going to bed with a good book. Or a friend who's read one." \(-\)Phyllis Diller In how many ways can these six jokes be ranked from best to worst?

Use the formula for \(_{n} C_{r}\) to solve There are 14 standbys who hope to get seats on a flight, but only 6 seats are available on the plane. How many different ways can the 6 people be selected?

Use the formula for \(_{n} C_{r}\) to solve An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the combinations formula to determine how many different four-note sound sequences can be created from the notes \(\mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G}, \mathrm{A},\) and \(\mathrm{B}\)

Exercises 86-88 will help you prepare for the material covered in the next section. $$\text { Evaluate } \frac{n !}{(n-r) !} \text { for } n-20 \text { and } r-3$$.
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