# Permutations and Combinations Worksheets

When we are determining the probability of a series of events there are situations where we need to focus on the order in which the events took place and other where this is irrelevant. This complicates things immensely when you are trying to determine the outcome of something that may have a substantial impact on your current situation. The first thing you need to do is decipher the relevance of the scenario that you are evaluating. If the order of events does matter it is a permutation. The math behind this follows a certain direction. If order does not matter at all, then it tracks as a combination that requires a whole different path of calculations. These worksheets and lessons help students learn how to understand and solve combination and permutation problems.

### Aligned Standard: HSS-CP.B.9

- Bag of Candy Step-by-step Lesson- 16 candidates, 3 positions on the tennis club board. How many probabilities are there?
- Guided Lesson - Arranging water bottles, seven colors of the rainbow, and grabbing cricket bats.
- Guided Lesson Explanation - It is also helpful to draw images along with the values.
- Practice Worksheet - We use the same formula to solve all the problems.
- Matching Worksheet - Match the problems to the number of possible combinations that you see.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

We look at the possible number of candidates and outcomes for different situations.

- Homework 1 - How many different combinations of management can there be to fill the positions of principal and vice principal of a school knowing that there are 11 eligible candidates?
- Homework 2 - John has 10 marbles in his bag. In how many ways can he pick 6 marbles from the bag?
- Homework 3 - Paul has 3 mobile phones in his desk. In how many different orders can the mobile phones be arranged?

### Practice Worksheets

Arrange letters in a word are the most common questions we have seen.

- Practice 1 - How many ways can the letters of the word PARK be arranged?
- Practice 2 - Maria has to visit 6 different places. In how many different ways can she visit them?
- Practice 3 - 8 different books are on a shelf. In how many different ways could you arrange them?

### Math Skill Quizzes

Make sure to speak to students on the possible arrangement of letters in words that have two of the same letter.

- Quiz 1 - How many 4 digit numbers can we make using the digits 3, 6, 7 and 8 without repetitions?
- Quiz 2 - Kelly has 5 water bottles in her desk. She numbers each bottle. How many different ways can she order her water bottles?
- Quiz 3 - How many ways can 7 people be arranged around a roundtable?

### What are Permutations and Combinations?

In statistics we will often come across situations that entail a series or cadre of events. When it comes to defining the order of objects or occurrence of events, there are two terms that we use in mathematics, and these are permutations and combinations. Out of these two, combination is the most used term and not many people think about "order" when they use this term. When the order does not matter, we use combinations and when changing the order will affect the outcome, we use permutations. When working on the combinations or permutations, there are two categories, one where repetition is allowed and the other where repetition is not allowed.

**Permutations** - The easiest of the two is one where repetition is allowed. All you need for the calculation such type of permutations is the number of choices available and the number of choices you must make. We can calculate permutations without repetition using the formula; n^{r}. Here; n is the total number of options or choices that are available, and r is the number of selections or choices that you have to make.

The second category is calculating permutations without repetition. In this case, we have to reduce the number of choices by one every time we make a selection. To calculate such permutations, we use the factorial function. When we to find out the ways some objects can be placed, without repetition, we use; n! = n × (n-1) × (n-2).

When we have to select r number of options from the total available options we use; n!/((n-r)!)

**Combinations** - When talking about calculating combinations without repetition, assume a situation where the order does matter and then alter it so the order does not matter. We can use the permutation formula of without reputation that involves a factorial function. As the order does not matter, we can alter the function; n!/(r!(n-r)!)

To determine the number of combinations with repetition, we can use the following formula; (r+n-1)!/r!(n-1)!

### When Would You Find the Need to Use These Skills in the Real World?

We live in a global corporate society that is obsessed with analytics and having the ability to analyze data to make well thought out decisions. Having the ability to not only perform these skills but recognizing the need for them in an applied situation is unbelievably valuable to many employers. When we want to gauge the scope of decision, comprehending how many diverse outcomes exist is key. This can be achieved by identifying all the possible variables that exist and compounding them into a calculation of the potential outcomes. Understanding combinations helps us gauge all the results in many different areas of interest. This includes but is not limited to the areas of technology (cryptography, data mining, and network communications), molecular biology (DNA and molecular interactions), pattern analysis (stock and equity moves). When the order in which data is layered matters, we are analyzing permutations. This skill is used to evaluate tournament schedules to make equitable match ups between opponents. The goal being to put players with as close to as even skills in head to head competition. This can be applied to any situation where we are concerned on the order in which the data lands in.