Question

You want to purchase vegetable dip for a party. A grocery store sells 7 different flavors of vegetable dip. You have enough money to purchase 2 flavors. How many combinations of 2 flavors of vegetable dip are possible?

Short answer

The number of possible combinations of 2 flavors of vegetable dip from a total of 7 is 21.

Step-by-step solution

Identify the values for n and r

First, we identify the values for \( n \) and \( r \) from the problem. Here, \( n \) (the total number of dips) is 7 and \( r \) (the number of dips to choose) is 2.

Use the combination equation to calculate

Next, we substitute the values of \( n \) and \( r \) into the combination equation. So, we will have: \( C(7,2) = 7! / [(7-2)!*2!] \).

Simplify the equation

Now, let's simplify this equation. This gives: \( C(7,2) = 7! / (5!*2!) \).

Calculate factorial values and solve the equation

Next, calculate the factorial values and solve the equation. This gives: \( C(7,2) = (7*6*5*4*3*2*1) / ((5*4*3*2*1)*(2*1)) = 21 combinations of 2 flavors \).

Key concepts

Factorials

Factorials are a fundamental concept in combinatorics and help solve problems related to counting and arrangements. The factorial of a non-negative integer, denoted by an exclamation mark (!), is the product of all positive integers less than or equal to that number. For instance, the factorial of 5 is:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials are used extensively in permutations and combinations, where they help determine how many ways items can be arranged or selected.
In this exercise, calculating factorials is essential to finding the number of combinations of vegetable dips. By understanding factorials, you gain insight into how different elements can be combined in various scenarios.

Combination Formula

The combination formula determines how many ways you can select items from a larger set when order does not matter. The formula is represented as:

• \( C(n, r) = \frac{n!}{(n-r)! \, r!} \)

Here, \( n \) represents the total number of items, and \( r \) represents the number of items to choose. This exercise involves choosing 2 flavors of vegetable dip from 7 available options:

• \( C(7, 2) = \frac{7!}{(7-2)! \, 2!} \)
• Simplified: \( \frac{7!}{5! \, 2!} \)

By substituting factorial values, we see that there are 21 ways to choose 2 flavors. Understanding the combination formula allows you to solve problems involving selection without concern for the order of items.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally distinct and separate. It encompasses topics like graphs, integers, and statements in logic. Combinatorics, which includes the study of counting and arrangement, is a key area in discrete math.
In the exercise, the problem of selecting combinations of vegetable dips is an example of discrete mathematics. It shows how specific choices or selections are made among distinct items. This field is essential in computer science, cryptography, and more, because it deals with finite systems rather than continuous ones.
Through exercises and applications, discrete mathematics provides tools to understand and solve practical counting and arrangement problems. It's all about understanding how things can be grouped or structured in different ways, an insight that's valuable in many real-life scenarios.