JEE Main 2026 — Jan 23, Shift 1 — Mathematics solution

JEE Main2026Jan 23, Shift 1Mathematics

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 $\frac{m}{n}$ , $g cd (m, n) = 1$ , then $m + n$ is equal to ______.

Your answer

Solution

Total ordered selections without replacement: $100 \times 99 = 9900$ .

Favourable. For each $b \in {1, 2, \dots, 90}$ , the number of valid $a$ values (with $a \geq b + 10$ ) is $100 - (b + 10) + 1 = 91 - b$ . Summing:

$\sum_{b = 1}^{90} (91 - b) = 90 + 89 + \dots + 1 = \frac{90 \cdot 91}{2} = 4095$ .

Probability $= \frac{4095}{9900} = \frac{91}{220}$ . Since $g cd (91, 220) = 1$ , $m = 91$ , $n = 220$ .

$m + n = 311$ .

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