Given there are k5 balls that we need to put into 3 buckets

Given there are k=5 balls that we need to put into 3 buckets. How many different results are there if the buckets and balls all have different labels? I think I have an idea for the answer but am not certain since I think it would be probably 3⁵ but that seems too large. I did try doing it by hand by labeling the buckets 1, 2, and 3 and the balls A,B, C, D, and E.

Solution

N THE FIRST BUCKET WE CAN CHOOSE 1 BALL OF 5 BALLS

IN THE SECOND BUCKET WE CAN CHOOSE 1 BALL OF 4 BALLS

IN THE 3 BUCKET WE CAN CHOOSE 1 BALL OF 3 BALLS

DIFFERENT WAYS = 5*4*3 =60

60 DIFFERENT WAYS AND we also have buckets are diffrent lables

Then it would be 60*3 = 180 diffrent combinations