/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 81 Solve for \(n\). $$_{n+1} P_{3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 81

Solve for \(n\). $$_{n+1} P_{3}=4 \cdot_{n} P_{2}$$

Short Answer

Expert verified
Therefore, the solution to the equation is \( n = 3 \).

Step by step solution

01

Rewrite the equations using the formula for permutations

In this step, we rewrite the given equations using the formula for permutations. So, \(_{n+1}P_{3} = \frac{(n+1)!}{((n+1)-3)!}\) and \(4 \cdot_{n}P_{2} = 4 \cdot \frac{n!}{(n-2)!}\).
02

Simplify equations

Next, we simplify the above equations. Therefore, \(_{n+1}P_{3} = \frac{(n+1)!}{(n-2)!}\) and \(4 \cdot_{n}P_{2} = 4 \cdot \frac{n!}{(n-2)!}\).
03

Equate both expressions of the permutation

At this stage, we are to equate both expressions of the permutation: \(\frac{(n+1)!}{(n-2)!} = 4 \cdot \frac{n!}{(n-2)!}\).
04

Cancel shared term

Since \((n-2)!\) appears in both sides of the equation, we can cancel it out: \( (n+1)! = 4 \cdot n! \).
05

Express the factorials and simplify

Express the factorials: \( (n+1) \cdot n! = 4 \cdot n! \). This simplifies to \( (n+1) = 4 \).
06

Solve for 'n'

Finally, we solve for \(n\) by subtracting 1 from both sides of the equation: \(n = 4 -1 \)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the binomial coefficient. \(\left(\begin{array}{c}100 \\ 98\end{array}\right)\)

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 0.) $$a_{n}=\frac{(-1)^{2 n}}{(2 n) !}$$

Simplify the factorial expression. $$\frac{12 !}{4 ! \cdot 8 !}$$

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$3,12,48,192, \ldots$$

Write the first five terms of the sequence defined recursively. $$a_{1}=15, a_{k}=a_{k-1}+3$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.