From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to a b , where a,b N and (a,b)=1 , then a+b is equal to ______

Step-by-step solution

The total number of ways to select 3 different dates from a month of 31 days is given by choosing 3 days out of 31 : ³¹C₃ = 31 30 29 3 2 1 = 4495 Let the three selected dates be x, y, z such that x 2y = x + z For y to be an integer, the sum x + z must be an even number. This is only possible if both x and z are either odd or even. Once x and z are chosen, y is uniquely determined as their exact average. In a month of 31 days, the number of odd dates ( 1, 3, 5, , 31 ) is 16 , and the number of even dates ( 2, 4, 6,