From the first 100 natural numbers, two numbers first a and then b are selected randomly without replacement. If the probability that is , then is equal to _______ . [2026]

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Solution

Q.
From the first 100 natural numbers, two numbers first a and then b are selected randomly without replacement. If the probability that $a - b \geq 10$ is $\frac{m}{n}$ , $\gcd (m, n) = 1,$ then $m + n$ is equal to _______ . [2026]

Ans.
(311)

$a - b \geq 10$

$Total cases = 100 \times 99$

$Fav. cases = 1 + 2 + 3 + \dots + 90$

$Req. Prob = \frac{1 + 2 + \dots + 90}{100 \times 99}$

$\frac{m}{n} = \frac{90 (\frac{91}{2})}{100 (99)} = \frac{91}{220}$

$m + n = 311$

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