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Chapter 5: Problem 30

List all the combinations of four objects \(a, b, c,\) and \(d\) taken two at a time. What is \(_{4} C_{2} ?\)

Short Answer

Expert verified
There are 6 combinations: \(a, b\), \(a, c\), \(a, d\), \(b, c\), \(b, d\), \(c, d\). \(_{4} C_{2} = 6\).

Step by step solution

01

- Understand the Problem

The problem requires finding all combinations of four objects taken two at a time. Additionally, compute the binomial coefficient \(_{4} C_{2}\).
02

- List All Combinations

To find all the combinations of the objects \(a, b, c, d\) taken two at a time, list pairs without repetition or consideration of order:1. \(a, b\)2. \(a, c\)3. \(a, d\)4. \(b, c\)5. \(b, d\)6. \(c, d\)
03

- Calculate \(_{4} C_{2}\) using the Formula

The binomial coefficient \(_{n} C_{r}\) is given by \(_{n} C_{r} = \frac{n!}{r!(n-r)!}\). Substituting \(n = 4\) and \(r = 2\): \(_{4} C_{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{24}{4} = 6\).

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Key Concepts

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

Binomial Coefficient
The binomial coefficient, often denoted by \({n \choose r}\), is a fundamental concept in combinatorics. It represents the number of ways to choose \(r\) objects from a set of \(n\) objects without regard to the order of selection. Understanding the binomial coefficient is essential for solving problems involving combinations. For example, in our exercise, identifying that \({4 \choose 2}\) calculates the number of ways to pick 2 items from a set of 4.
Combinations
Combinations refer to the selection of items from a larger pool where the order does not matter. To find combinations, you can use the formula for the binomial coefficient. Let's revisit the exercise where we list combinations of four objects \(a, b, c, d\) taken two at a time.

Here are the pairs:
- \(a, b\)
- \(a, c\)
- \(a, d\)
- \(b, c\)
- \(b, d\)
- \(c, d\)

This systematically covers all possible pairs without any repeated elements.
Factorials
Factorials are a way to multiply a series of descending natural numbers. Symbolized by \(!\), they are used heavily in combinatorics problems. For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). This concept helps in calculating the binomial coefficient as it is used in the formula:

\[ _{n} C_{r}= \frac{n!}{r!(n-r)!} \]
This helps break down complex problems into manageable steps, as seen when calculating \(_{4} C_{2}\).
Discrete Mathematics
Discrete mathematics involves studying mathematical structures that are fundamentally discrete rather than continuous. Combinatorics, a branch of discrete mathematics, focuses on counting, arrangement, and combination of sets of elements. It's crucial for solving problems like the exercise given, where we analyze distinct pairs from a finite set. Mastering discrete mathematics provides a strong foundation for computer science, cryptography, and algorithm design.

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Most popular questions from this chapter

The following data represent the number of driver fatalities in the United States in 2002 by age for male and female drivers: $$\begin{array}{|l|c|c|} \hline \text { Age} & \text { Male } & \text { Female }\\\\\hline \text { Under } 16 & 228 & 108 \\\\\hline 16-20 & 5696 & 2386 \\\\\hline 21-34 & 13,553 & 4148 \\\\\hline 35-54 & 14,395 & 5017 \\\\\hline 55-69 & 4937 & 1708 \\\\\hline 70 \text { and over } & 3159 & 1529 \\\\\hline\end{array}$$ (a) What is the probability that a randomly selected driver fatality who was male was 16 to 20 years old? (b) What is the probability that a randomly selected driver fatality who was 16 to 20 was male? (c) Suppose you are a police officer called to the scene of a traffic accident with a fatality. The dispatcher states that the victim is 16 to 20 years old, but the gender is not known. Is the victim more likely to be male or female? Why?

Suppose a compact disk (CD) you just purchased has 13 tracks. After listening to the CD, you decide that you like 5 of the songs. The random feature on your CD player will play each of the 13 songs once in a random order. Find the probability that among the first 4 songs played (a) you like 2 of them; (b) you like 3 of them; (c) you like all 4 of them.

In a recent basketball game, a player who makes \(65 \%\) of his free throws made eight consecutive free throws. Assuming free-throw shots are independent, determine whether this feat was unusual.

In a survey of 500 randomly selected high school students, it was determined that 288 played organized sports. What is the probability that a randomly selected high school student plays organized sports?

A woman has five blouses and three skirts. Assuming that they all match, how many different outfits can she wear?
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