There are apparently $3072$ ways to draw this flower. But why? This picture was in my friend's math book: 

 

Below the picture it says: 


  There are $3072$ ways to draw this flower, starting from the center of
  the petals, without lifting the pen.


I know it's based on combinatorics, but I don't know how to show that there are actually $3072$ ways to do this. I'd be glad if someone showed how to show that there are exactly $3072$ ways to draw this flower, starting from the center of the petals, without lifting the pen (assuming that $3072$ is the correct amount).
First you have to draw the petals.  There are $4!=24$ ways to choose the order of the petals and $2^4=16$ ways to choose the direction you go around each petal.  Then you go down the stem to the leaves.  There are $2! \cdot 2^2=8$ ways to draw the leaves.  Finally you draw the lower stem.  $24 \cdot 16 \cdot 8=3072$
The presumption is wrong!

Most answers here start like this:
"First you have to draw the petals."  

This is a mistake. You could start drawing the flower from two points: The centerpoint of the blossom OR the endpoint at the lower stem. 

Result is:
There are 2 x 3072 = 6144 ways to draw the Flower! 

I think it might have to do with the way you asked the question. ;)