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

Or every three spices (no the isn"t greedy),

**Or** you could say "none at every thanks", i m sorry is the "empty set":

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

Has the subsets:

alexbillyetc ...You are watching: How many subsets does a set of n elements have

It likewise has the subsets:

alex, billyalex, caseybilly, daleetc ...Also:

alex, billy, caseyalex, billy, daleetc ...And also:

the whole set: alex, billy, casey, dalethe empty set:Now let"s begin with the Empty collection and move on increase ...

## TheEmpty Set

How countless subsets does the empty collection have?

You can choose:

the whole set: the empty set:

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

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

It is choose asking "There is nothing available, for this reason what execute you choose?" prize "nothing". The is your just choice. Done.

## ASet with One Element

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

apple

How countless **subsets** walk the collection apple have?

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

So any collection with **one** element will have actually **2** subsets.

## ASet v Two Elements

Let"s add another element to our example set:

apple, banana

How numerous subsets walk the collection apple, banana have?

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

**apple, banana**the empty set:

So a collection with **two** elements has **4** subsets.

## ASet With three Elements

How about:

apple, banana, cherry

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

Subsets v one element: **apple**, **banana**, **cherry**

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

And:

the whole set:**apple, banana, cherry**the north set:

In truth we can put the in a table:

List | Number of 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: did you view a sample in the numbers there?)

## Setswith Four aspects (Your Turn!)

Now try to perform the same for this set:

apple, banana, cherry, date

Here is a table for you:

List | Number the subsets | |

zero elements | ||

one element | ||

two elements | ||

three elements | ||

four elements | ||

Total: |

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

## Setswith five Elements

And now:

apple, banana, cherry, date, egg

Here is a table for 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 6 Elements

What about:

apple, banana, cherry, date, egg, fudge

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

How countless subsets space there because that a collection of 6 elements? _____How countless subsets space there because that a collection of 7 elements? _____

## AnotherPattern

Now let"s think around subsets and sizes:

Theemptyset hasjust**1subset**: 1A set with one facet has

**1 subset**with no elements and

**1subset**v one element: 1 1A collection with twoelements has actually

**1 subset**v no elements,

**2 subsets**v one element and also

**1 subset**with two elements: 12 1A collection with threeelements has actually

**1 subset**with no elements,

**3 subsets**with oneelement,

**3 subsets**with two elements and also

**1 subset**through threeelements: 1 3 3 1and therefore on!

Do you identify thispattern that numbers?

They are the number from Pascal"sTriangle!

This is **very useful**, since now friend can inspect if you have the right variety of subsets.

Note: the rows start at 0, and an in similar way the columns.

Example: because that the set **apple, banana, cherry, date, egg** you perform subsets of length three:

But the is only **4** subsets, how numerous should over there be?

Well, friend are selecting 3 the end of 5, so go to **row 5, place 3** the Pascal"s Triangle (remember to start counting at 0) to uncover you require **10 subsets**, for this reason you must 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 over there a means of calculating the numbers such as **1, 4, 6, 4 and also 1** (instead of feather them up in Pascal"s Triangle)?

Yes, we can discover the variety of ways of choosing each number ofelements utilizing Combinations.

There space four aspects in the set, and:

The number of ways ofselecting 0 aspects from 4 = 4C0 =

**1**The number of ways ofselecting 1 element from 4 = 4C1 =

**4**The variety of ways of picking 2 aspects from 4 = 4C2 =

**6**The variety of ways of choosing 3 facets from 4 = 4C3 =

**4**The number of ways of choosing 4 aspects from 4 = 4C4 =

**1**full number ofsubsets =

**16**

The variety of waysofselecting 0 aspects from 5 = 5C0 = 1The variety of ways ofselecting 1 facet from 5 = ___________The number of ways of selecting 2 facets from 5 = ___________The variety of ways of choosing 3 facets from 5 = ___________The variety of ways of choosing 4 aspects from 5 = ___________Thenumber of ways of picking 5 aspects from 5 = ___________ Total variety of subsets = ___________

## Conclusion

In this activity you have:

Discovered a ascendancy fordetermining the total variety of subsets for a provided set: A set with nelements has actually 2n subsets.Found a connection betweenthe number of subsets that each dimension with the numbers in Pascal"striangle.Discovered a quick method tocalculate this numbers using Combinations.See more: Is Heating Sand A Chemical Change ? Is Sand To Glass A Chemical Or A Physical Change

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