/*! 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 90 In Exercises 87-90, prove the id... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 90

In Exercises 87-90, prove the identity. $$_{n} C_{r}=\frac{n P_{r}}{r !}$$

Short Answer

Expert verified
The given identity \( _{n} C_{r}=\frac{n P_{r}}{r !} \) is verified and proven to be true.

Step by step solution

01

Write down the formulas for combination and permutation.

The formula for combination is \( _{n} C_{r} = \frac{n !}{r ! (n-r) !} \) and the formula for permutation is \( n P_{r} = \frac{n !}{(n-r) !} \)
02

Substitute the formulas into the identity.

Substituting the formulas into the identity \( _{n} C_{r}=\frac{n P_{r}}{r !} \) to be proven, we get \( \frac{n !}{r ! (n-r) !} = \frac{\frac{n !}{(n-r) !}}{r !} \)
03

Simplify the equation.

Simplify the equation by multiplying both sides by \( r ! (n-r) ! \), it gives \( n ! = n ! \) which is true
04

Verify the identity.

Since both sides of the equation are equal, the identity is proven to be true

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.

Permutations
In combinatorics, a permutation is a way to arrange a set of objects in an ordered sequence. Think of it like arranging books on a shelf. The order in which you place them matters.
For a given number of objects, the formula to calculate the number of possible permutations is denoted by \( nP_r \). The formula for permutations is:
\[nP_r = \frac{n!}{(n-r)!}\]In this equation:
  • \(n\) is the total number of items;
  • \(r\) is the number of items being arranged or selected;
  • \(!\) (factorial) represents the product of an integer and all the positive integers below it 鈥� for example, \(4! = 4 \times 3 \times 2 \times 1 = 24\).
Permutations are useful in situations where the order is crucial. For example, determining the various ways to assign runners in a race would involve permutations.
Combinations
Combinations, unlike permutations, are selections of items where the order doesn't matter. It's like choosing a group of friends to invite over for a game night, where the order in which you invite them does not affect the group.
The formula for combinations is represented as \(_{n}C_{r}\) and can be calculated as:
\[_{n}C_{r} = \frac{n!}{r!(n-r)!}\]Here's what each part represents:
  • \(n!\) is the factorial of the total number of items;
  • \(r!\) is the factorial of the number of selected items;
  • \((n-r)!\) is the factorial of the difference between the total items and the selected items.
Combinations are used when the grouping of items is important, but the arrangement is not. A typical example is choosing a team from a list of candidates.
Mathematical Proof
A mathematical proof is a logical argument that demonstrates the truth of a given statement. Proving identities, such as the one in the exercise, is about showing that two expressions are equal under certain conditions.
In proving \(_{n}C_{r} = \frac{nP_{r}}{r!}\), we follow these steps:
  • Substitute the known formulas: Insert the formulas for permutations and combinations into the given identity.
  • Perform algebraic manipulations: Simplify the equation step-by-step, like multiplying both sides by a common factor to reveal it's true.
  • Verify the expressions: Ensure both sides of the equation simplify to the same expression, confirming the identity.
Proofs help us to establish truths in mathematics by methodically aligning each step to show definitive results. They enable us to trust that what we are stating is universally true, regardless of particular values for \(n\) and \(r\).

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

Finding a Sum In Exercises \(45-54\) , find the sum using the formulas for the sums of powers of integers. $$\sum_{n=1}^{20}\left(n^{3}-n\right)$$

Finding a Sum In Exercises \(45-54\) , find the sum using the formulas for the sums of powers of integers. $$\sum_{i=1}^{6}\left(6 i-8 i^{3}\right)$$

Assume that the probability of the birth of a child of a particular sex is 50\(\% .\) In a family with four children, what is the probability that (a) all the children are boys, (b) all the children are the same sex, and (c) there is at least one boy?

Finding a Linear or Quadratic Model In Exercises \(55-60\) , decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model. $$-2,1,6,13,22,33, \dots$$

Linear Model, Quadratic Model, or Neither? In Exercises \(61-68\) , write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither. $$a_{1}=2$$ $$a_{n}=a_{n-1}+2$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.