Given an unlimited supply of balls of n different colors and

Given an unlimited supply of balls of n different colors and a prime number p. show that the number of different arrangements of p balls in a circle is exactly (n^p - n) / p if we exclude the arrangements in which all the balls are the same color. Specially address why p must be prime. This proves that p divides n^p - n which is Fermat\'s little theorem

Solution

Given that there are n different colors and number of balls is p.

So, the number of different arrangements of p balls in n different colours is equal to n^p.

when we exclude the arrangements in which all the balls are the same color is n

the number of different arrangements of p balls in a circle

(n^p-n)/p