Problem 28 - Algebra 2 and Trigonometry

Question

In how many ways can 5 new bus passengers choose seats if there are 8 empty seats?

Step-by-Step Solution

Answer

There are 56 ways for 5 passengers to choose seats from 8 empty seats.

Step 1: Identify the Problem

We need to determine the number of ways 5 passengers can choose seats from 8 available empty seats.

Step 2: Use the Combinatorial Formula

To find how many ways 5 passengers can choose 5 seats out of 8, we use the combination formula: \( C(n, r) = \frac{n!}{r!(n-r)!} \) where \( n \) is the total seats and \( r \) is the number of seats chosen.

Step 3: Insert Known Values into the Formula

Here, \( n = 8 \) and \( r = 5 \), so we substitute these values into the formula: \[ C(8, 5) = \frac{8!}{5!(8-5)!} = \frac{8!}{5! \times 3!} \]

Step 4: Calculate Factorials

Calculate the factorials involved: - \( 8! = 40320 \) - \( 5! = 120 \) - \( 3! = 6 \).

Step 5: Simplify the Expression

Substitute the factorials into the formula: \[ C(8, 5) = \frac{40320}{120 \times 6} = \frac{40320}{720} \]

Step 6: Solve the Division

Perform the division: \( \frac{40320}{720} = 56 \).

Step 7: Conclusion

The total number of ways the passengers can choose the seats is 56.

Key Concepts

Combinations

Combinations are a fundamental concept in combinatorics, which is a field of mathematics dealing with the counting, arrangement, and combination of objects. In mathematics, a combination refers to a selection of items from a larger pool where the order of selection does not matter. This contrasts with permutations, where the order does matter.
Understanding combinations is crucial when you need to know how many different groups can be formed out of a larger set without worrying about the order of selection. For instance, if you want to choose a committee of 3 people from a group of 10, you would use combinations to figure out how many different committees could be formed.
Mathematically, combinations are calculated using the binomial coefficient formula, which is often represented as \( C(n, r) \), where \( n \) is the total number of items, and \( r \) is the number of items to choose. The formula is \( C(n, r) = \frac{n!}{r!(n-r)!} \). The exclamation mark denotes a factorial, which is another key concept we will explore next.

Factorials

Factorials are a mathematical operation that is foundational in understanding permutations, combinations, and many other aspects of combinatorics. The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \).

• For example, \( 5! \) is calculated as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• Similarly, \( 0! \) is defined to be 1 by convention.

Factorials grow very rapidly with increasing \( n \) and are used primarily in the calculations of permutations and combinations.
The rapid growth of factorial values makes manual calculations challenging for large numbers, but appreciated tools like calculators or software can handle these computations smoothly, making factorials quite accessible in practical applications.

Permutations

Permutations are another key aspect of combinatorics, where the order of items is significant. If you're arranging a set of items into a sequence, permutations will tell you how many possible arrangements, or orders, you can create with those items.
For instance, if you wanted to arrange 3 books on a shelf, and the order matters, you would calculate the permutations to see the total number of possible arrangements. Using factorials once again, permutations are calculated with the formula, \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items and \( r \) is the number of items being arranged.
Consider we have 3 books and we need to arrange all 3 of them. The calculation is straightforward:

• First, find 3! which is \( 3 \times 2 \times 1 = 6 \).
• So, there are 6 permutations if we are to arrange all 3 books.

In cases like these, permutations provide a way to understand how small changes in order can lead to multiple unique arrangements.

Binomial Coefficient

The binomial coefficient is a central concept in combinatorics, critical for calculating combinations. It is often represented in mathematical expressions using the notation \( \binom{n}{r} \), pronounced "\( n \) choose \( r \)." This component directly connects to the combination formula, \( C(n, r) = \frac{n!}{r!(n-r)!} \), which allows us to determine how many ways we can select \( r \) items from a larger group of \( n \) items, without considering the order of selection.
Binomial coefficients also play a significant role in the expansion of binomial expressions. For example, in the expansion of \((a+b)^n\), the coefficients are determined by these binomial coefficients and demonstrate their prevalence beyond just selecting groups from a set.
These coefficients are valuable in various problems not only in mathematics but also in computer science, physics, and other fields where determining combinations of elements is essential.