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Chapter 11: Problem 75

I used the formula for \({ }_{n} C_{r}\) to determine how many outcomes are possible when choosing four letters from a, \(\mathrm{d}, \mathrm{h}, \mathrm{n}, \mathrm{p}\), and \(w .\)

Short Answer

Expert verified
There are 15 possible ways to choose four letters from a, d, h, n, p, and w.

Step by step solution

01

Identify 'n' and 'r' values

In the combination formula \({ }_{n} C_{r}\), 'n' represents the total number of options and 'r' represents the number of options being chosen at once. For this problem, there are six total letters (a, d, h, n, p, w) so 'n' is 6. We are choosing four letters, so 'r' is 4.
02

Use the combination formula

The combination formula is given by \({ }_{n} C_{r} = \frac{n!}{r!(n-r)!}\). '!' denotes factorial, which means multiplying all positive integers from our chosen number down to 1. Next, substitute the values of 'n' and 'r' into the combination formula. This gives us \({ }_{6} C_{4} = \frac{6!}{4!(6-4)!}\).
03

Calculate the factorial values

Calculate the values for the factorials in our equation: 6! = 6*5*4*3*2*1, 4! = 4*3*2*1, and (6-4)! which is 2! = 2*1.
04

Substitute the factorial values into the formula

After calculating the factorial values, substitute them into the equation in Step 2. The equation becomes \({ }_{6} C_{4} = \frac{6*5*4*3*2*1}{4*3*2*1*2*1}\).
05

Simplify the equation

Cancel out the common terms in the numerator and the denominator to get \({ }_{6} C_{4} = \frac{6*5}{2*1}\). Final calculation yields \({ }_{6} C_{4} = 15\).

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Key Concepts

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

Factorial
In mathematics, the concept of a factorial is a way to express the product of all positive integers up to a given number. For instance, when we write a factorial, it is symbolized by the exclamation mark (!). So for a number like 5, the factorial is represented as 5! and calculated as follows:
  • 5! = 5 脳 4 脳 3 脳 2 脳 1 = 120
This sequence of multiplication starts from the number itself and goes down to 1. Factorials are applied in many areas of mathematics, especially in permutations and combinations.
This powerful tool helps us compute arrangements and selections, making it easier to tackle problems involving grouping different objects.
Combinatorics
Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It's a fundamental part of discrete mathematics used to solve real-world problems involving configurations.
  • Combinatorics helps us understand how different sets of elements can be chosen and arranged.
  • It plays a critical role in computer science, optimization, algorithm design, and more.

By understanding the underlying principles of combinatorics, such as the concept of counting and arrangement, you can solve complex problems by breaking them down into simpler, smaller steps. This field is essential for managing large datasets and making calculations involving permutations and combinations more manageable.
Permutations and Combinations
Permutations and combinations are concepts in combinatorics that describe different ways of arranging and selecting objects. Though the terms might sound similar, they refer to different ideas.
  • Permutations: This describes arrangements where the order of selection matters. For example, if you are arranging books on a shelf, each order is a different permutation. Formula: \( P(n, r) = \frac{n!}{(n-r)!} \)
  • Combinations: Unlike permutations, combinations focus on selections where the order does not matter, such as choosing four letters from a set of six, where each group of letters is considered equal. Formula: \( C(n, r) = \frac{n!}{r!(n-r)!} \)
These concepts are vital in probability and statistics, helping estimate different scenarios without needing to count each one individually. They provide a way to explore how groups behave under different conditions, assisting in predictions and decision-making processes.
Mathematical Formulas
Mathematical formulas are expressions representing relationships between different quantities, crucial for solving problems efficiently. They offer a systematic method for calculating values and understanding patterns in data.
  • A formula provides a quick way to compute an answer, reducing complex problems to simple calculations.
  • In combinatorics, the formula for combinations, \( C(n, r) = \frac{n!}{r!(n-r)!} \), is used to find how many ways you can choose 'r' elements from 'n' without regard to order.

Understanding how formulas work allows you to predict outcomes and create solutions for diverse mathematical challenges. They represent the bedrock of algebra, calculus, physics, and many other scientific disciplines, enabling us to shift from abstract thinking to practical application seamlessly.

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Most popular questions from this chapter

This activity is a group research project intended for people interested in games of chance at casinos. The research should culminate in a seminar on games of chance and their expected values. The seminar is intended to last about 30 minutes and should result in an interesting and informative presentation made to the entire class. Each member of the group should research a game available at a typical casino. Describe the game to the class and compute its expected value. After each member has done this, so that class members now have an idea of those games with the greatest and smallest house advantages, a final group member might want to research and present ways for currently treating people whose addiction to these games has caused their lives to swirl out of control.

Involve computing expected values in games of chance. Another option in a roulette game (see Example 6 on page 753 ) is to bet \(\$ 1\) on red. (There are 18 red compartments, 18 black compartments, and 2 compartments that are neither red nor black.) If the ball lands on red, you get to keep the \(\$ 1\) that you paid to play the game and you are awarded \(\$ 1\). If the ball lands elsewhere, you are awarded nothing and the \(\$ 1\) that you bet is collected. Find the expected value for playing roulette if you bet \(\$ 1\) on red. Describe what this number means.

It is estimated that there are 27 deaths for every 10 million people who use airplanes. A company that sells flight insurance provides \(\$ 100,000\) in case of death in a plane crash. A policy can be purchased for \(\$ 1\). Calculate the expected value and thereby determine how much the insurance company can make over the long run for each policy that it sells.

How many different four-letter radio station call letters can be formed if the first letter must be W or K?

The tables in Exercises 3-4 show claims and their probabilities for an insurance company. a. Calculate the expected value and describe what this means in practical terms. b. How much should the company charge as an average premium so that it breaks even on its claim costs? c. How much should the company charge to make a profit of \(\$ 50\) per policy? PROBABILITIES FOR HOMEOWNERS' INSURANCE CLAIMS $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Amount of Claim (to the } \\ \text { nearest } \$ \mathbf{\$ 5 0 , 0 0 0 )} \end{array} & \text { Probability } \\ \hline \$ 0 & 0.65 \\ \hline \$ 50,000 & 0.20 \\ \hline \$ 100,000 & 0.10 \\ \hline \$ 150,000 & 0.03 \\ \hline \$ 200,000 & 0.01 \\ \hline \$ 250,000 & 0.01 \\ \hline \end{array} $$
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