Two distinct integers are chosen at random from the first 20 natural numbers. Given that the sum of the two numbers is even, the probability that the absolute difference between the two numbers is at least 10 is :

Step-by-step solution

Let the chosen numbers be x and y from 1, 2, , 20 . Since their sum is even, both numbers must be even or both must be odd. Number of ways to choose two even numbers = ¹⁰C₂ = 45 . Number of ways to choose two odd numbers = ¹⁰C₂ = 45 . Total number of outcomes in the reduced sample space = 45 + 45 = 90 . We need the probability that |x - y| 10 . Since x and y have the same parity, their absolute difference must be an even integer. Thus, |x - y| can be 10, 12, 14, 16, or 18 . Number of pairs with difference 10 : (1,