It has 4 elements.

When making a subset, there are:

2 choices to make concerning the element w, (1) to include w or 
(2) not to include w.

For each of those 2 choices to make concerning the element w, there are
2 choices to make concerning the element x, (1) to include x or 
(2) not to include x.  That's 2x2 or 4 choices to make concerning w and x.

For each of those 4 choices to make concerning the elements w and x, there are
2 choices to make concerning the element y, (1) to include y or 
(2) not to include y.  That's 2x2x2 or 8 choices to make concerning w, x, and y.

For each of those 8 choices to make concerning the elements w, x, and x, there
are 2 choices to make concerning the element z, (1) to include z or 
(2) not to include z.  That's 2x2x2x2 or 16 choices to make concerning w, x, y,
and z.

so the set {w,x,y,z} has 24 = 16 subsets, including the empty set and
the whole set.

They are:

 1. { }
 2. {w}
 3. {x}
 4. {y}
 5. {z}
 6. {w,x}
 7. {w,y}
 8. {w,z}
 9. (x,y}
10. {x,z}
11. {y,z}
12. {w,x,y}
13. (w,x,z}
14. {w,y,z}
15. (x,y,z}
16. {w,x,y,z}

Edwin