Question

how many mappings are there from an nth element set onto a two element set?

Answer 1

Is this a multiple choice answer?

Answer 2

no it isn't

Answer 3

You can derive it by looking at some concrete examples.

How many functions from {a} to {0,1}
We can list the mappings

{(a,0)}
{(a,1)}

So there are 2 mappings from a set of 1 element to a set of 2 elements.

How many mappings from
{a,b} to { 0,1}

{(a,0),(b,0)}
{(a,0),(b,1)}
{(a,1),(b,0)}
{(a,1),(b,1)}

So there are 4 mappings from a set of 2 elements to a set of 2 elements.

How many mappings from
{a,b,c} -> { 0,1}

{(a,0),(b,0),(c,0)}

{(a,0),(b,0),(c,1)}
...

It is getting tedious to write out all the mappings.
But it turns out that the number of mappings from an n element set to {0,1} is equivalent to the number of strings of length n made from elements 0 and 1 (also called binary strings).

So for example the mappings from {a,b} to {0,1} can be rewritten as strings 00, 01, 10 ,11

The number of mappings from {a,b,c} to {0,1} can be rewritten as
000, 001, 010, 011, 100, 101, 110, 111

for a string of length n , strings can be formed using 0 and 1?
Well use the multiplication principle. for the first entry there are 2 choices 0 or 1 , for the second entry there are also 2 choices of 0 or 1. Keep going until the last entry.
so there are 2x2x2x...2 strings of length n

Can you answer the question now?