Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

An ice cream parlor stocks 31 different flavors and advertises that it serves almost 4500 different triple scoop cones, with each scoop being a different flavor. How was this number obtained?

Step-by-Step Solution

Answer

The number of different triple scoop cones is 4495.

Step 1: Understanding the Problem

The ice cream parlor offers 31 different flavors. We need to determine how many unique combinations of triple scoop cones can be created where each scoop consists of a different flavor.

Step 2: Using Combinations Without Repetition

Since each of the three scoops must have a different flavor, and the order in which the flavors are scooped doesn't matter, we use combinations without repetition. The combination formula is \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items chosen. Here \( n = 31 \) and \( r = 3 \).

Step 3: Applying the Combination Formula

Calculate \( \binom{31}{3} \) to find the number of different triple scoop cones possible. The formula is given by \( \binom{31}{3} = \frac{31 \times 30 \times 29}{3 \times 2 \times 1} \).

Step 4: Calculating the Numerator and Denominator

First, calculate the product of the numbers in the numerator: \( 31 \times 30 \times 29 = 26970 \). Then calculate the denominator: \( 3 \times 2 \times 1 = 6 \).

Step 5: Final Calculation

Divide the numerator by the denominator to find the number of combinations: \( \frac{26970}{6} = 4495 \). Therefore, there are 4495 different triple scoop combinations.

Key Concepts

Combinations Without Repetition

In the world of combinatorics, one vital concept is determining how many ways you can choose items from a group without repeating any of them. This is known as "combinations without repetition." It involves selecting a subset where the order doesn't matter, and each item is picked only once. Imagine you have several unique items, and you wish to pick a few of them, but you are not bothered about the sequence in which you pick them.
Here's a practical example: Suppose you are at an ice cream parlor with 31 unique flavors and want to enjoy a triple scoop cone with each scoop being a different flavor. In this scenario, the sequence in which you stack your ice creams doesn't matter – just that each scoop is different.
This is a typical example of a combination without repetition, making it crucial to understand this concept when faced with selection problems where order is unimportant, such as forming a team, selecting toppings, or making unique sets.

Combination Formula

The combination formula is an essential tool for calculating combinations without repetition. It helps determine how many ways we can choose a given number of items from a larger collection, where the order of selection does not matter. The formula is represented as \( \binom{n}{r} \), which reads as "n choose r." Here, \( n \) is the total number of items, and \( r \) is the number of items you pick.
The formula itself is expressed mathematically as:

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

In simpler terms, it involves factorials of numbers, where factorial \( n! \) is the product of all positive integers from 1 to \( n \). In our ice cream parlor example with 31 flavors, if you want to find out how many different triple scoop cones can be created, you'll apply this formula where \( n = 31 \) and \( r = 3 \). This calculation simplifies to:

• \( \binom{31}{3} = \frac{31 \times 30 \times 29}{3 \times 2 \times 1} \)

This formula and calculation give the exact number of unique ways to arrange your scoops without repetition and regardless of order.

Mathematical Problem Solving

Mathematical problem-solving often requires understanding and applying the right formulas and concepts. It's essential to break down problems into smaller steps and apply logical reasoning, especially in combinatorial problems like our triple scoop ice cream cone example.
Here’s an organized approach you might use in solving such problems:

• **Understand the Problem:** Clearly define what is being asked. Here, we need to find the number of different combinations of triple scoops from 31 flavors.
• **Identify the Right Concept:** Recognize that the situation requires using combinations without repetition because the order doesn't matter, and each flavor is only used once.
• **Apply the Correct Formula:** Use the combination formula \( \binom{n}{r} \) to calculate the total combinations possible, often simplifying it by calculating parts of the formula separately (like numerator and denominator).
• **Perform Calculations:** Execute the mathematical operations to achieve the final count. For instance, \( 26970 \div 6 \), resulting in 4495 distinct ways.

By following these steps, you can efficiently tackle combinatorial problems, leading to accurate and insightful results.