If the total number of ways of selecting two numbers from the set 1 , 2 , 3 , … … … . . , 89 , 90 such that their sum is divisible by 3 is k , then k 500 is

Step-by-step solution

Divide the numbers in 3 sets of the form 3 k , 3 k + 1 and 3 k + 2 S 1 = 1 , 4 , 7 , … … , 88 S 2 = 2 , 5 , 8 , … … . , 89 S 3 = 3,6 , 9 , … … , 90 For the sum of two numbers to be divisible by three, either we will choose the two numbers from S 3 or one number from S 1 and one from S 2 i.e. k = C 1 30 × C 1 30 + C 2 30 = 30 × 30 + 30 × 29 2 = 1335 Hence, 1335 500 = 2 . 67