could you clarify this a bit

Do you just need adjacent colors to be different or do you need to see all 5 colors in your set of 5 spins?

Never mind I see you need all 5 colors.

Any permutation of all 5 colors is allowed. There are $5!$ of these.

Each individual spin has a probability of $\dfrac 1 {3125}$

Thus your probability is

$Pr[$all 5 colors in 5 spins$]=\dfrac {5!}{3125}=\dfrac{120}{3125}=\dfrac {24}{625}$