First we see how many ways we can have the outer two digits. 
They can be

1 _ _ 1

8 _ _ 8

6 _ _ 9

9 _ _ 6

So we can pick the outer digits 4 different ways.

Next we see how many ways we can have the middle two digits. 
They can be

_ 0 0 _

_ 1 1 _

_ 8 8 _

_ 6 9 _

_ 9 6 _

So we can pick middle two digits 5 different ways. 


So for each of the 4 ways we can pick the outer two digit,
we can pick the inner two digits 5 ways.
So there are 4×5 or 20 ways to have a palidum.

Here are all 20 palidia:

 1. 1 0 0 1
 2. 8 0 0 8
 3. 6 0 0 9
 4. 9 0 0 6
 5. 1 1 1 1
 6. 8 1 1 8
 7. 6 1 1 9
 8. 9 1 1 6
 9. 1 8 8 1
10. 8 8 8 8
11. 6 8 8 9
12. 9 8 8 6
13. 1 6 9 1
14. 8 6 9 8
15. 6 6 9 9
16. 9 6 9 6
17. 1 9 6 1
18. 8 9 6 8
19. 6 9 6 9
20. 9 9 6 6 

Edwin