Problem 44 - College Algebra

Question

Use the formula for \(_{n} C\), to solve Exercises \(41-48\). There are 14 standbys who hope to get seats on a flight, but only 6 seats are available on the plane. How many different ways can the 6 people be selected?

Step-by-Step Solution

Answer

The number of different ways in which the 6 people can be selected from a group of 14 is given by the combination \( C(14, 6) \).

Step 1: Identify the given variables

In this exercise, the number of possible standbys, which we'll call \( n \), is 14. The number of seats available, which we'll call \( r \), is 6.

Step 2: Apply the Formula for Combinations

Now, apply the formula for combinations \( C(n, r) = \frac{n!}{r!(n-r)!} \). Substituting the values of \( n \) and \( r \) into this formula, we get \( C(14, 6) = \frac{14!}{6!(14-6)!} \).

Step 3: Calculate Factorials

Compute factorials of 14, 6, and 8. Factorial of a number \( n \) represented as \( n! \) is the product of all positive integers less than or equal to \( n \). Make sure to calculate these before proceeding to the next step.

Step 4: Complete the Calculation

Finally, divide the calculated factorial of 14 by the multiplication of the calculated factorial of 6 and 8 as per the combination formula: \( C(14, 6) = \frac{14!}{6!*8!} \). Calculate the right-hand side of this equation to find the value of \( C(14, 6) \).

Key Concepts

Factorials

The concept of factorials is a fundamental building block in combinatorial mathematics. A factorial, denoted as \( n! \), is the product of all positive integers from 1 to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow very rapidly with larger numbers.
This operation is used to determine the number of ways in which objects can be arranged, which is essential in solving problems involving permutations and combinations.

• Zero factorial, \( 0! \), is defined as 1. This is mainly because there is exactly one way to arrange zero objects: to do nothing.
• A factorial can also be broken down using smaller factorials, for instance, \( 14! = 14 \times 13! \).

Knowing how to calculate factorials is crucial because they are used in the formula for the binomial coefficient, which we'll explore in the next section.

Binomial Coefficient

The binomial coefficient, expressed as \( C(n, r) \), represents the number of ways to choose \( r \) objects from a set of \( n \) objects without considering the order. It is synonymous with the term "combinations."
The formula for this coefficient is \( \frac{n!}{r!(n-r)!} \). This formula subtracts the arrangements of unused elements, yielding the different groupings possible for the specified size.

• In our exercise, we're using \( C(14, 6) \) to find the number of ways to select 6 individuals from 14 available standbys.
  
• This involves calculating the factorials: \( 14! \), \( 6! \), and \( (14-6)! \).

The binomial coefficient plays a crucial role in probability and statistics; it helps with calculations where the order does not matter. For example, if you're dealt a hand of cards, it's often the combination of cards, not the order, that counts.

Combinatorial Mathematics

Combinatorial mathematics is a field that studies the counting, arrangement, and combination of objects. It has practical applications in computer science, geometry, and optimization tasks.
When tackling combinatorial problems, the focus is often on figuring out the number of ways to arrange or select items under certain conditions, which is exactly what we aimed to address in the exercise.

• Combinatorial analysis asks questions like, "How many ways can a group of items be selected?" or "What is the probability of a specific outcome?"
• It also investigates complex topics like permutations (where order matters) and combinations (where order does not matter).

This exercise that asks how to choose 6 people from a group of 14 is a classic example of combinatorics, demonstrating how mathematical principles can solve real-world problems like seating arrangements or planning tasks. With the understanding of factorials and the binomial coefficient, solving such problems becomes systematic and achievable.