Problem 49 - College Algebra

Question

A motorcycle shop has 10 choppers, 6 bobbers, and 5 café racers-different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 café racers for a weekend showcase?

Step-by-Step Solution

Answer

There are 7,200 ways to choose the motorcycles for the showcase.

Step 1: Identify the Problem

The problem asks for the number of ways to choose specific quantities of different types of motorcycles from given totals. This is a combination problem because the order of selection does not matter.

Step 2: Calculate Chopper Combinations

Use the combination formula \( \binom{n}{k} \), where \( n \) is the total number of choppers and \( k \) is the number of choppers to choose. For choppers, \( n = 10 \) and \( k = 3 \), so the formula is: \[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120. \]

Step 3: Calculate Bobber Combinations

For bobbers, \( n = 6 \) and \( k = 5 \). Use the combination formula: \[ \binom{6}{5} = \frac{6!}{5!(6-5)!} = \frac{6}{1} = 6. \]

Step 4: Calculate Café Racer Combinations

For café racers, \( n = 5 \) and \( k = 2 \). Use the combination formula: \[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10. \]

Step 5: Compute Total Combinations

Multiply the number of combinations for each motorcycle type together to find the total number of ways to choose the motorcycles. Thus, the total ways is: \[ 120 \times 6 \times 10 = 7200. \]

Key Concepts

Combination formula

The combination formula is a fundamental concept in combinatorics used to determine the number of ways to select a subset of items from a larger set when the order of selection does not matter. It is represented mathematically as \( \binom{n}{k} \), which is read as "n choose k". This formula is applicable when you want to choose \( k \) items from \( n \) items without caring about the sequence of the selection.

To compute this, you use the formula:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \). The factorial function plays a key role in calculating combinations, as it allows the reduction of repeated elements in both the chosen set and those not chosen in the subset.

In our motorcycle example, this formula helps calculate how many different ways the shop can choose specific numbers of choppers, bobbers, and café racers for the showcase, with each choice being independent of the others.

Factorial

Factorials are a mathematical operation that multiplies a series of descending natural numbers. Represented as \( n! \), where \( n \) is a non-negative integer, the factorial function is pivotal in permutations and combinations because it provides a systematic way to count arrangements and selections.

For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 0! = 1 \) by convention, to simplify combinatorial formulas

The factorial is the backbone of the combination formula, because it helps to arrange the total number of items \( n \) and specifically account for the \( k \) items being chosen (and not chosen). Understanding factorials is crucial for applying permutation and combination principles and calculating specific scenarios such as the number of choppers chosen in the motorcycle shop problem.

Permutation and combination

Permutations and combinations are central concepts in combinatorics that help count the arrangements and selections of items. While permutations consider the arrangement order, combinations do not, making them ideal for the motorcycle problem.

• Permutations focus on arrangements: \( n \) items to fill \( r \) spots use \( nPr = \frac{n!}{(n-r)!} \).
• Combinations focus on selections: \( n \) items into \( r \) groups use \( \binom{n}{r} \).

The problem from the motorcycle shop is solved using combinations because the order of choosing motorcycles doesn't matter for planning the showcase.Both methods, permutations, and combinations, involve factorial operations but differentiate by their treatment of item order. Recognizing when to implement permutations or combinations depends on whether the arrangement order impacts the outcome, which in this context, it does not.

Counting principles

Counting principles provide various strategies to determine the total number of possible outcomes in a given problem. A key principle used in the motorcycle exercise is the multiplication principle, which allows calculations of total outcomes by breaking the situation into sequential events.

In this problem, each type of motorcycle (chopper, bobber, café racer) is treated as a separate event. By calculating the combinations for each separately and then multiplying them together:

• Choices for choppers \( = 120 \)
• Choices for bobbers \( = 6 \)
• Choices for café racers \( = 10 \)

Total ways \( = 120 \times 6 \times 10 = 7200 \)

This multiplication principle is a fundamental counting strategy in combinatorics, useful for deriving totals from independent scenarios. It extensively aids in situations where multiple steps lead to an end goal, such as in our shop planning a showcase.