Or any type of two flavors: banana, chocolate, banana, vanilla, or chocolate, vanilla,

Or all three seasonings (no the isn"t greedy),

Or you might say "none at all thanks", which is the "empty set":

### Example: The set alex, billy, casey, dale

Has the subsets:

alexbillyetc ...

You are watching: How many subsets in a set with 4 elements

alex, billyalex, caseybilly, daleetc ...

Also:

alex, billy, caseyalex, billy, daleetc ...

And also:

the totality set: alex, billy, casey, dalethe empty set:

Now let"s start with the Empty collection and move on increase ...

## TheEmpty Set

How numerous subsets go the empty set have?

You might choose:

the entirety set: the north set:

But, hang on a minute, in this instance those are the same thing!

So theempty set really has just 1 subset (whichis itself, the north set).

It is prefer asking "There is nothing available, therefore what execute you choose?" answer "nothing". The is your only choice. Done.

## ASet with One Element

The set could be anything, but let"s just say the is:

apple

How plenty of subsets does the collection apple have?

the entirety set: applethe empty set:

And that"s all.Youcanchoose the one element, or nothing.

So any set with one element will have actually 2 subsets.

## ASet v Two Elements

Let"s include another aspect to our instance set:

apple, banana

How many subsets go the set apple, banana have?

It can have apple, or banana, and don"t forget:

the whole set: apple, bananathe empty set:

So a collection with two facets has 4 subsets.

## ASet With three Elements

apple, banana, cherry

OK, let"s be an ext systematic now, and list the subsets by exactly how many facets they have:

Subsets with one element: apple, banana, cherry

Subsets with two elements: apple, banana, apple, cherry, banana, cherry

And:

the totality set: apple, banana, cherrythe empty set:

In reality we can put that in a table:

 List Number that subsets zero elements 1 one element apple, banana, cherry 3 two elements apple, banana, apple, cherry, banana, cherry 3 three elements apple, banana, cherry 1 Total: 8

(Note: go you see a sample in the numbers there?)

## Setswith Four aspects (Your Turn!)

Now shot to perform the exact same for this set:

apple, banana, cherry, date

Here is a table because that you:

 List Number of subsets zero elements one element two elements three elements four elements Total:

(Note: if friend did this right, there will certainly be a pattern to the numbers.)

## Setswith 5 Elements

And now:

apple, banana, cherry, date, egg

Here is a table because that you:

 List Number the subsets zero elements one element two elements three elements four elements five elements Total:

(Was over there a sample to the numbers?)

## Setswith six Elements

apple, banana, cherry, date, egg, fudge

OK ... Us don"t need to finish a table, because...

How plenty of subsets room there because that a collection of 6 elements? _____How many subsets space there because that a collection of 7 elements? _____

## AnotherPattern

Now let"s think about subsets and also sizes:

Theemptyset hasjust 1subset: 1A set with one facet has 1 subset v no elements and also 1subset through one element: 1 1A collection with twoelements has 1 subset v no elements, 2 subsets with one element and also 1 subset v two elements: 12 1A set with threeelements has 1 subset v no elements, 3 subsets through oneelement, 3 subsets through two elements and 1 subset with threeelements: 1 3 3 1and for this reason on!

Do you recognize thispattern of numbers?

They room the number from Pascal"sTriangle!

This is very useful, due to the fact that now friend can inspect if you have actually the right variety of subsets.

Note: the rows start at 0, and likewise the columns.

Example: for the set apple, banana, cherry, date, egg you perform subsets of size three:

apple, banana, cherryapple, banana, dateapple, banana, eggapple, cherry, egg

But that is just 4 subsets, how numerous should over there be?

Well, friend are choosing 3 out of 5, so go to row 5, position 3 the Pascal"s Triangle (remember to start counting in ~ 0) to find you need 10 subsets, so you need to think harder!

In fact these are the results: apple,banana,cherry apple,banana,date apple,banana,egg apple,cherry,date apple,cherry,egg apple,date,egg banana,cherry,date banana,cherry,egg banana,date,egg cherry,date,egg

## Calculating The Numbers

Is there a means of calculating the numbers such as 1, 4, 6, 4 and also 1 (instead of looking them increase in Pascal"s Triangle)?

Yes, us can find the number of ways of choosing each number ofelements using Combinations.

There are four facets in the set, and:

The variety of ways ofselecting 0 elements from 4 = 4C0 = 1The number of ways ofselecting 1 element from 4 = 4C1 = 4The variety of ways of choosing 2 aspects from 4 = 4C2 = 6The number of ways of choosing 3 facets from 4 = 4C3 = 4The number of ways of picking 4 aspects from 4 = 4C4 = 1 total number ofsubsets = 16
The number of waysofselecting 0 aspects from 5 = 5C0 = 1The number of ways ofselecting 1 facet from 5 = ___________The variety of ways of choosing 2 facets from 5 = ___________The number of ways of picking 3 facets from 5 = ___________The number of ways of picking 4 aspects from 5 = ___________Thenumber of methods of selecting 5 aspects from 5 = ___________ Total variety of subsets = ___________

## Conclusion

In this activity you have:

Discovered a dominance fordetermining the total variety of subsets for a given set: A set with nelements has actually 2n subsets.Found a connection betweenthe number of subsets of each size with the number in Pascal"striangle.Discovered a quick method tocalculate this numbers utilizing Combinations.

See more: How Many Side Does A Cube Have, What Is A Cube

Moreimportantly you have actually learned how different branches of mathematics canbe merged together.