Combinations and Permutations

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From the course by University System of Georgia

Six Sigma Advanced Define and Measure Phases

83 оценки

University System of Georgia

Six Sigma Advanced Define and Measure Phases

83 оценки

Course 2 of 4 in the Specialization Six Sigma Green Belt

This course is for you if you are looking to dive deeper into Six Sigma or strengthen and expand your knowledge of the basic components of green belt level of Six Sigma and Lean. Six Sigma skills are widely sought by employers both nationally and internationally. These skills have been proven to help improve business processes and performance. This course will take you deeper into the principles and tools associated with the "Design" and "Measure" phases of the DMAIC structure of Six Sigma. It is highly recommended that you complete the "Yellow Belt Specialization" and the course "Six Sigma and the Organization (Advanced)" before beginning this course. In this course, your instructors will introduce you to, and have you apply, some of the tools and metrics that are critical components of Six Sigma. This course will provide you with the advanced knowledge of team dynamics and performance, process analysis, probability, statistics, statistical distributions, collecting and summarizing data, measurement systems analysis, process and performance capability, and exploratory data analysis associated with Six Sigma and Lean. Every module will include readings, videos, and quizzes to help make sure you understand the material and concepts that are studied. Registration includes online access to course content, projects, and resources but does not include the companion text The Certified Six Sigma Green Belt Handbook (2nd edition). The companion text is NOT required to complete the assignments. However, the text is a recognized handbook used by professionals in the field. Also, it is a highly-recommended text for those wishing to move forward in Six Sigma and eventually gain certification from professional agencies such as American Society for Quality (ASQ).

From the lesson

Probability and Statistics - pt2

In this module, you will learn how to solve probability problems related to combinations and permutations. You will also learn about central limit theorem and how to apply it to the concept of confidence intervals. This module also touches on descriptive and inferential statistics.

Combinations and Permutations4:59

The Central Limit Theorem (CLT)5:24

The Central Limit Theorem (CLT) Applications3:02

Descriptive vs. Inferential Statistics2:22

Meet the Instructors

Christina Scherrer, PhD
Professor of Industrial Engineering
Department of Systems and Industrial Engineering
Gregory Wiles, PhD
Interim Chair and Assistant Professor
Department of Systems and Industrial Engineering
Bill Bailey, PhD
Assistant Professor of Industrial Engineering
Department of Systems and Industrial Engineering

In the previous module we learned about various probability rules.

In this module, we will start by covering the last of the probability rules

permutations and combinations.

I'll be honest, people usually either love or hate these topics.

A lot of fun probability problems are based in these two rules,

but they also can be tricky.

Let's dive in.

A permutation is an ordered arrangement of distinct objects.

The number of ways of ordering the arrangement of r objects taken from

a set of n objects is abbreviated as nPr.

In permutations, the fact that we are ordering the items

rather than taking them as an unordered group matters.

The formula for nPr is n!/(n- r)!.

Most scientific calculators have an nPr key.

And you can also use Excel with the function; =Permut(n,r).

For example, suppose with your newfound knowledge of probability,

you've managed to win the lottery and are celebrating by hiring a private yacht to

sail you and your closest friends around the Caribbean for a week.

How many seven-day itineraries are possible?

If you plan to visit one different island nation per day, and

there are 28 island nations to choose from.

Here we have 28 islands to choose from, so that is our n.

We're going to visit 7 of them, so that is our r.

And in an itinerary, the order that you visit the islands matters.

So, permutations is appropriate.

That leaves us with 28 P7,

which is calculated 28!/(28-7)!.

Alternatively, you could type in

=PERMUT(28,7) in an empty cell in Excel.

That comes out to 5,967,561,600, almost 6 billion.

Permutations grow very quickly.

Whichever of those itineraries you chose though, I'd be happy to go along.

Now, what if you weren't interested in the order of the itinerary, but

rather, just which seven islands you ended up visiting overall?

Permutations would no longer be the tool to use.

For that, we would need to use combinations.

With combinations, we are just taking a group of items and

not putting them in order.

The abbreviation is nCr, and

the formula is nCr equals n!/[r!(n-r)!].

Again, there's often a calculator function, or

you can use =Combin(n,r) in Excel.

So, with the island yachting

adventure we would have 28 C7,

which is 28!/[7!(28-7)!],

or =Combin(28,7) in Excel.

That value is 1,184,040.

That is still a large number, but the values for combinations will be

significantly smaller than permutations because we don't care about the order.

With both permutations and

combinations we are taking a subset of size r from the whole group n.

The difference is that we use permutations when order matters and

combinations when order does not matter.

One trick to keep the two straight is to think Permutations = Places.

Both start with the letter P, and

when running a race, you care what place you finish in.

Being first is different than being third,

even though both end up on the medal stand.

And then, Combinations = Committees, both start with C.

Whether your chosen first or third to be on a committee makes no difference.

You still need to do the same amount of work.

Let's look at a few more examples.

If you're hosting a dinner party and randomly choosing which wine to pair with

each of the 5 courses you're serving from the 35 varieties in your wine cellar,

would you use permutations or combinations?

That would be permutations because whether a wine is paired with your salad course or

your dessert course is different, the order matters.

On the other hand, if you're hosting a cocktail party instead and

just putting out five types of wine on the counter for guests to help themselves,

then the order the wines are chosen by you doesn't matter anymore.

And we would use combinations.

What about if you've received a lot of 100 items from your supplier and

are going to randomly select 5 of them for

quality testing to determine if you will accept the lot?

If you find more than one defect, you will reject the lot.

Does the order you sample the items matter?

No, it doesn't.

Whether the defects are found toward the beginning or the end, you will reject

the lot if there are two or more defects, so we would use combinations.

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