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How many possibilities does a navigation have with 5 different items.

I first though of 5x5 but that is not enough I think.

For example, these are some possibilities:

The letters represent simple navigation text items like Home, Users, Login ...

A

B

C

D

E

A     B     C     D     E

A     B     C     D

A     B     C 

A     B  

A   
...

if I have for example this:

A    B

and

B    A

it is redundant in my case, the order is not important.

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    I don't understand what a "navigation with n items" represents. – 2017-01-04
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    just buttons or text items like Home | Users | Tags ...., I shortened them with letters in my example – 2017-01-04
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    And the question is about combinations containing from 1 to 5 of those 5 items? – 2017-01-04
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    Your question is as clear as the skies over Beijing on a bad day. – 2017-01-04
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    @gunbl4d3 yes it is about the combination – 2017-01-04
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    By navigation, can I assume $A, B, C...$ to be independent unit vectors in different directions. And the no. of ways to find new vectors by finding sum of any or all of these vectors. – 2017-01-04
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    It sounds like you are asking how many non-empty subsets exist for a five element set. As there are $2^n$ subsets for an $n$ element set including the empty subset, there are $2^5-1=31$ subsets here. – 2017-01-04

2 Answers 2

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The number of ways to select $m$ elements from a set of $N$ elements is given by the combinatorial number $N \choose m$.

So the number of combinations of 1 to 5 elements from a set of 5 elements is:

$$\sum_{i=1}^5 {5 \choose i}=31$$

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The number of different locations you can reach with $n$ navigation directions is

$$^nC_1+ ^nC_2+...+^nC_n=(1+1)^n-1=2^n-1$$