Problem 43 - Algebra 2

Question

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities. picking 3 apples from the last 7 remaining at the grocery store

Step-by-Step Solution

Answer

This is a combination scenario, with 35 possible ways to pick the apples.

Step 1: Determine the Scenario Type

First, determine whether the situation involves a permutation or a combination. In this case, we are choosing 3 apples out of a total of 7. Since the order in which we pick the apples does not matter, this scenario involves a combination.

Step 2: Define the Combination Formula

The formula for a combination is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 3: Identify Values for Formula

Here, \( n = 7 \) and \( r = 3 \) because there are 7 apples to choose from, and we want to select 3 of them.

Step 4: Plug Values into the Formula

Substitute the values \( n = 7 \) and \( r = 3 \) into the combination formula: \[ C(7, 3) = \frac{7!}{3!(7-3)!} \]

Step 5: Calculate Factorials

Compute the factorials: \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \), \( 3! = 3 \times 2 \times 1 = 6 \), and \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Step 6: Simplify and Calculate

Substitute these factorial values back into the combination formula: \[ C(7, 3) = \frac{5040}{6 \times 24} \]. Simplify to get \[ C(7, 3) = \frac{5040}{144} = 35 \].

Key Concepts

Factorials

Factorials are a fundamental part of many mathematical equations, especially in combinatorics, permutations, and combinations. A factorial is the product of an integer and all the integers below it. For instance, the factorial of 5, denoted as 5!, is equal to 5 × 4 × 3 × 2 × 1, which equals 120. Factorials are used to calculate the number of ways to arrange a set of elements. The concept of factorials helps to determine possibilities as it provides a way to express the grown relationship between numbers related to sequences or arrangements.

When you encounter a problem involving arrangements, sequences, or groupings, think of factorials as your go-to tool. They simplify computation and help easily solve complex permutation and combination problems when you plug them into formulas.

Permutations vs Combinations

Understanding the difference between permutations and combinations is crucial in probability and statistics. Permutations consider the order of selection to be important. For example, if you are arranging 3 books on a shelf from a selection of 7 books, different orders count separately.

The permutations formula is expressed as:

• Permutations (nPr): \( nPr = \frac{n!}{(n-r)!} \)

In contrast, combinations ignore the order of selection. The scenario of picking 3 apples from 7 at a store is a combination because it doesn't matter which apple is chosen first. Each different grouping counts only once.

The combinations formula is expressed as:

• Combinations (nCr): \( nCr = \frac{n!}{r!(n-r)!} \)

Therefore, when solving problems like determining how many ways you can pick items, ascertain whether the order matters to decide between permutations and combinations.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and the counting of possible arrangements. It applies to different fields like computer science, cryptography, and genetics where counting distinct arrangements and selections matters.

A key idea in combinatorics is not just to count, but to count only the feasible options. Using factorials, permutations, and combinations, you can systematically approach problems involving arrangement and selection. This branch greatly benefits anyone solving problems related to selecting, arranging, or forming groups. Combinatorics problems may include real-world applications like determining how to assign committees, how students can be seated in a class, or ways to arrange digits in number systems.

Probability Theory

Probability theory is the mathematical foundation for analyzing random phenomena and is often intertwined with combinatorics. Probability involves calculating the likelihood of different outcomes. While combinatorics gives a way to count possible outcomes, probability goes a step further to study how likely each outcome is to occur.

In simple terms, probability is expressed as a fraction of the number of favored outcomes over the number of total possible outcomes. For example, if you want to know the probability of drawing a specific type of apple from a basket containing various apples, you use combinations to figure out the total number of possible choices and then find your probability accordingly. These principles of probability theory help in making informed predictions in everyday life, from predicting weather patterns to deciding risks in business.