Or any two flavors: banana, chocolate, banana, vanilla, or cacao, vanilla,

Or all three flavors (no that isn"t greedy),

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


Example: The collection alex, billy, casey, dale

Has the subsets:

alexbillyand so on ...

You are watching: How many subsets of three elements each exist in a set of six elements?

It additionally has actually the subsets:

alex, billyalex, caseybilly, daleetc ...

Also:

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

And also:

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

Now let"s begin with the Empty Set and move on up ...

TheEmpty Set

How many type of subsets does the empty collection have?

You can choose:

the entirety set: the empty set:

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

So theempty set really has simply 1 subset (whichis itself, the empty set).

It is prefer asking "There is nopoint obtainable, so what carry out you choose?" Answer "nothing". That is your only option. Done.

ASet With One Element

The collection might be anything, however let"s simply say it is:

apple

How many type of subsets does the set apple have?

the whole set: applethe empty set:

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

So any collection via one aspect will certainly have 2 subsets.

ASet With Two Elements

Let"s add another element to our instance set:

apple, banana

How many type of subsets does the set apple, banana have?

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

the entirety set: apple, bananathe empty set:

So a set via two facets has 4 subsets.

ASet With Three Elements

How about:

apple, banana, cherry

OK, let"s be more systematic now, and list the subsets by exactly how many type of elements they have:

Subsets with one element: apple, banana, cherry

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

And:

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

In reality we can put it in a table:

ListNumber of subsets
zero elements1
one elementapple, banana, cherry 3
two elementsapple, banana, apple, cherry, banana, cherry3
three elementsapple, banana, cherry1
Total:8

(Note: did you see a pattern in the numbers there?)

Setswith Four Elements (Your Turn!)

Now try to perform the very same for this set:

apple, banana, cherry, date

Here is a table for you:

ListNumber of subsets
zero elements
one element
two elements
3 elements
4 elements
Total:

(Note: if you did this best, tright here will certainly be a pattern to the numbers.)

Setsthrough Five Elements

And now:

apple, banana, cherry, day, egg

Here is a table for you:

ListNumber of subsets
zero elements
one element
2 elements
3 elements
4 elements
five elements
Total:

(Was tbelow a pattern to the numbers?)

Setswith Six Elements

What about:

apple, banana, cherry, day, egg, fudge

OK ... we don"t should finish a table, because...


How many kind of subsets are tright here for a collection of 6 elements? _____How many kind of subsets are tright here for a collection of 7 elements? _____

AnotherPattern

Now let"s think about subsets and sizes:

Theemptyset hasjust 1subset: 1A set through one facet has actually 1 subset via no elements and 1subset via one element: 1 1A set with twoelements has 1 subset via no aspects, 2 subsets through one aspect and 1 subset with two elements: 12 1A set via threeelements has actually 1 subset through no aspects, 3 subsets with oneelement, 3 subsets through two aspects and also 1 subset with threeelements: 1 3 3 1and also so on!

Do you acknowledge thispattern of numbers?

They are the numbers from Pascal"sTriangle!


*

This is incredibly useful, because currently you deserve to check if you have actually the best number of subsets.

Note: the rows begin at 0, and additionally the columns.


Example: For the set apple, banana, cherry, date, egg you list subsets of size three:

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

But that is just 4 subsets, exactly how many need to there be?

Well, you are picking 3 out of 5, so go to row 5, place 3 of Pascal"s Triangle (remember to start counting at 0) to find you need 10 subsets, so you must think harder!

In reality 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,day,egg


Calculating The Numbers

Is tright here a method of calculating the numbers such as 1, 4, 6, 4 and also 1 (rather of looking them up in Pascal"s Triangle)?

Yes, we have the right to discover the variety of means of selecting each number ofelements using Combinations.

Tbelow are 4 facets in the collection, and:


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

Conclusion

In this activity you have:

Disspanned a preeminence fordetermining the full number of subsets for a provided set: A set through naspects has actually 2n subsets.Found a link betweenthe numbers of subsets of each size via the numbers in Pascal"striangle.Discovered a quick way tocalculate these numbers making use of Combicountries.

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Moreimportantly you have actually learned how various branches of math canbe combined together.