From the first 100 natural numbers, two numbers first a and then b are selected randomly without replacement. If the probability that a-b 10 is m n , gcd (m, n)=1 , then m+n is equal to _ _ _ _ .

Step-by-step solution

Total ordered selections = 100 99 = 9900 . Favorable ( a - b 10 ): for each b from 1 to 90, a ranges from b+10 to 100, giving 91 - b choices. Count = _ b=1 ⁹⁰(91-b) = _ k=1 ⁹⁰k = 90 91 2 = 4095 . Probability = 4095 9900 = 91 220 (after dividing by 45). (91, 220) = 1 , so m = 91, n = 220 . m + n = 311 .