Combinations

What are Combinations?

Combinations are a way to calculate the number of ways to choose a subset of items from a larger set, without considering the order of selection. In other words, combinations are used to count the number of ways to select a certain number of items from a larger group, where the order of selection does not matter.

Formula for Combinations

The formula to calculate the number of combinations of n items taken r at a time is given by:

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

Where:

n is the total number of items
r is the number of items to be chosen
! represents the factorial of a number

Example

For example, if you have 5 different colors of pens and you want to choose 2 of them, you can calculate the number of different pairs of pens you can choose using the combination formula:

\[C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10\]

So, there are 10 different combinations of 2 pens that can be chosen from 5 different colors.

Properties of Combinations

Some important properties of combinations include:

Combinations are unordered selections.
The number of combinations is always less than or equal to the number of permutations (where order matters).
Choosing 0 items from a set results in 1 combination (the empty set).
Choosing all items from a set results in 1 combination (the entire set itself).

Study Guide

When studying combinations, it's important to understand the concept of choosing items without considering the order of selection. You should also be familiar with the combination formula and how to apply it to different scenarios. Additionally, practice solving problems involving combinations to reinforce your understanding of the topic.

Here are some key points to focus on while studying combinations:

Understanding the difference between combinations and permutations.
Mastering the combination formula and its application in various problems.
Recognizing situations where combinations are used, such as in counting the number of ways to form a committee, select a team, or choose a subset of items.

By grasping these concepts and practicing problems related to combinations, you will be well-prepared to tackle any questions or exercises on this topic.

◂Math Worksheets and Study Guides Eighth Grade. Displaying data

Organizing & Displaying Data

The resources above cover the following skills:

Data Analysis and Probability (NCTM)

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

Select and use appropriate statistical methods to analyze data.

Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

Develop and evaluate inferences and predictions that are based on data.

Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.

Grade 8 Curriculum Focal Points (NCTM)

Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets

Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose.

Connections to the Grade 8 Focal Points (NCTM)

Data Analysis: Building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatterplots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures.

Combinations

What are Combinations?

Formula for Combinations

Example

Properties of Combinations

Study Guide

[Combinations] Related Worksheets and Study Guides:

◂Math Worksheets and Study Guides Eighth Grade. Displaying data

The resources above cover the following skills: