Fermat's theorem states that, if p is a prime and x is any natural number, then either p divides x or is divisible by p. Hence, if x is a positive integer, then either 3 divides x or is divisible by 3. Similarly, either 3 divides y or is divisible 3. Observe that - ( ) is divisible by 3 if both x and y are not multiples of 3 .
Among , there are n multiples of 3 . Among the multiples of 3, we can select two one after other without replacement in ways. Out of the remaining 2 n numbers, we can select two one after other without replacement ways. Hence the required probability is