Problem 52 - Algebra and Trigonometry

Question

These problems involve combinations. Three-Topping Pizzas A pizza parlor offers a choice of 16 different toppings. How many three-topping pizzas are possible?

Step-by-Step Solution

Answer

There are 560 possible combinations of three-topping pizzas.

Step 1: Understanding the Combination Formula

In this problem, we need to determine how many combinations of 3 toppings can be made from a total of 16 toppings. This is a typical combination problem, which can be solved using the combination formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 2: Identify Parameters for the Problem

The parameters in this problem are \( n = 16 \) and \( r = 3 \) because we have 16 toppings to choose from, and we need to select 3 toppings for each pizza. We will use these values in our combination formula.

Step 3: Substitute Parameters into the Formula

Substitute \( n = 16 \) and \( r = 3 \) into the combination formula:\[C(16, 3) = \frac{16!}{3!(16-3)!}\]

Step 4: Simplify the Factorials

Simplify \( 16! \), \( 3! \), and \( 13! \) to find the number of combinations:- Calculate \( 16! = 16 \times 15 \times 14 \times 13! \)- Calculate \( 3! = 3 \times 2 \times 1 = 6 \)- Realize that \( \frac{16!}{13!} = 16 \times 15 \times 14 \)- Substitute these into the formula:\[C(16, 3) = \frac{16 \times 15 \times 14}{6}\]

Step 5: Calculate the Final Result

Continue to simplify:- First, calculate the numerator: \( 16 \times 15 \times 14 = 3360 \)- Then divide by 6: \( \frac{3360}{6} = 560 \)Thus, there are 560 possible combinations of three-topping pizzas.

Key Concepts

Factorials

Factorials are a special mathematical operation used to find the total number of ways to arrange a set of items.
It is denoted by the symbol ! (exclamation point). The factorial of a number is the product of all positive integers up to that number.

• For example, 5! (read as "five factorial") is equal to 5 × 4 × 3 × 2 × 1, which is 120.
• Factorials grow very large, very quickly. Even just 10! results in the number 3,628,800.

Factorials are key in calculating combinations and permutations since they allow us to determine the number of possible arrangements or groups.
In the pizza problem, calculating 16! involves multiplying all numbers from 16 down to 1, but in reality often simplifies to just a few steps; here, it is enough to directly multiply 16 down to 14 for our purposes.

Combination Formula

The combination formula is a fundamental tool in combinatorics, used to determine how many subsets of a particular size can be made from a larger set.
Combinations differ from permutations because the order of selection doesn’t matter.
The formula is written as:
\[C(n, r) = \frac{n!}{r!(n-r)!}\]where:

• \( n \) is the total number of items, and
• \( r \) is the number of items to choose.

For the pizza problem, we have 16 possible toppings and we need to select 3, so \( n = 16 \) and \( r = 3 \).
The combination formula helps us compute not just the possible pizza combinations, but any scenario where a subset needs to be formed from a larger set without regard to order.
This is why the formula subtracts arrangements involving order, simplifying complex problems effectively.

Counting Principles

Counting principles are the rules that guide our reasoning when determining the number of ways events can occur or selections can be made.
They form the basis of the more advanced combination problems.
The fundamental counting principle states that if there are multiple ways to do something, the total number of outcomes is the product of the number of ways each step can be completed.
In a combination problem like selecting pizza toppings, it involves:

• Selecting 3 toppings "together" rather than separate, without concern for sequence.
• Ensuring that each topping selection is distinct and doesn't repeat.
• Using combinations specifically because, unlike permutations, order does not change the outcome's identity.

Counting principles ensure we accurately reflect reality in mathematical models, allowing us to move from straightforward tasks like sequence determination to more complex combination tasks.