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We have 5 options and must choose 4. However, the same color can be used more than once. 5*4 = 20, and  ${20}\choose{4}$ = 4845.

We did overcount (Example: 1234 is the same as 2341, but we counted it twice), so to take away overcounting, we divide 4845 by 5 to get 969. With some other casework and steps shown below, you will end up with $\boxed{165}$.

Do not try to list out the answers because there are many and that is inefficient. Always use number theory. 

- PM

All the same colour = 5 ways

3 the same and one different = 5C2*2 = 20 ways

2 of one colour and 2 of another = 5C2*2 =20ways

2 the same and 2 different = 5C3*3*3 = 90ways

all colours different = 5C4*3*2= 30 ways

5+30+20+20+90 = 165 ways