The circle is immaterial in this one, instead think of this question as how can you fill 4 boxes with 5 colours, with no 2 adjacent boxes with the same color. Here's a (bad) representation of the boxes

|5 |4 |4 |4 |

Now, first box can be filled in 5 ways, next one in 4 ways (since same colors cant be adjacent), next one again in 4 and same for the last one.

So, total number of ways = 5 x 4 x 4 x 4 = 320

dont u think it should be 5x4x4x3. at last quart we have 3 options because it has two adjacent quarts and colored used here can not be used

quantskillsgmat wrote:dont u think it should be 5x4x4x3. at last quart we have 3 options because it has two adjacent quarts and colored used here can not be used

Thats a good question, and made me think as well. But consider this, lets say we have 5 colours, ABCDE.
Now, imagine filling in the circle in a clockwise fashion. First spot obviously can be 5 ways, next one 4, the one after another 4. As for the final piece, we could still have 4 options, since repetition is allowed.

E.g, if you fill in A,B,A, then you still have 4 options to fill the final pie, since the 2 colours next to the final pie will be the same.