/*! 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 14 Evaluate the expression. $$_{1... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 14

Evaluate the expression. $$_{12} P_{0}$$

Short Answer

Expert verified
The permutation of 12 items taken 0 at a time, \(_{12}P_0\), equals 1.

Step by step solution

01

Understand the permutation formula

The permutation formula is given by \(_nP_r = \frac{n!}{(n-r)!}\), where n is the total number of items, r is the number of items taken at a time, and '!' denotes a factorial, which is the product of an integer and all the integers below it, i.e., n! = n*(n-1)*(n-2)*...*3*2*1. In this case, n = 12 and r = 0.
02

Substitute the values into the permutation formula

The substituted permutation formula becomes \(_{12}P_0 = \frac{12!}{(12-0)!}\). This simplifies to \(_{12}P_0 = \frac{12!}{12!}\).
03

Solve the formula

Since the numerator and the denominator are the same, the result of the permutation, \(_{12}P_0\), is 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!

Key Concepts

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

Factorial
The factorial of a number is a key concept in mathematics, especially in permutations and combinations. It is represented by an exclamation mark (!). For any positive integer \( n \), the factorial \( n! \) is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Here are a few important points about factorials:
  • \( 0! \) is defined to be 1. This is a crucial rule that may seem counterintuitive but simplifies many equations.
  • Factorials grow very quickly. Even for relatively small numbers, such as 10, the factorial can be quite large.
Factorials are foundational in exploring permutations since they allow us to calculate the number of ways to arrange a set of items.
Combinatorics
Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It helps us to solve problems related to permutations and combinations by providing structured methods and formulas.

Why is combinatorics important?
  • It helps in finding how many ways we can choose items or arrange them.
  • From puzzles to coding algorithms, combinatorics plays a significant role in practical applications like cryptography and data analysis.
In our exercise, we're dealing with permutations, a specific type of combinatorial arrangement where the order of arrangement matters.
Permutation Formula
The permutation formula is crucial for determining the number of ways to arrange a subset of items from a larger set. It is expressed as:
\[_nP_r = \frac{n!}{(n-r)!}\]where \( n \) is the number of total items, and \( r \) is the number of items to arrange.

Key points about the permutation formula:
  • When \( r = 0 \), which means no items are chosen, the formula simplifies to 1, because there is exactly one way to arrange nothing.
  • The formula accounts for different orders, which differentiates permutations from combinations where order doesn't matter.
In our example, \( _{12}P_0 \) evaluated to 1, illustrating that there's one way to arrange no items from twelve, reinforcing that the foundational concepts of permutations align with intuitive counting principles.

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

MODELING WITH MATHEMATICS A football team is losing by 14 points near the end of a game. The team scores two touchdowns (worth 6 points each) before the end of the game. After each touchdown, the coach must decide whether to go for I point with a kick (which is successful 99\(\%\) of the time) or 2 points with a run or pass (which is successful 45\(\%\) of the time). a. If the team goes for 1 point after each touchdown, what is the probability that the team wins? loses? ties? b. If the team goes for 2 points after each touchdown, what is the probability that the team wins? loses? ties? c. Can you develop a strategy so that the coach's team has a probability of winning the game that is greater than the probability of losing? If so, explain your strategy and calculate the probabilities of winning and losing the game.

In Exercises 13 and \(14,\) you roll a six-sided die. Find \(P(A \) or \(B) .\) Event \(A :\) Roll a \(6 .\) Event \(B :\) Roll a prime number.

You make 6 posters to hold up at a basketball game. Each poster has a letter of the word TIGERS. You and 5 friends sit next to each other in a row. The posters are distributed at random. Find the probability that TIGERS is spelled correctly when you hold up the posters.

Simplify the expression. Write your answer using only positive exponents. $$\frac{6 y^{0}}{3 x^{-6}}$$

Evaluate the expression. $$_{30} P_{2}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.