JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Probability

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

\frac{m}{n}, gcd (m, n) = 1

, then

m + n

is equal to ______.

Answer: 311

Solution

Total ways to select a then b without replacement:

100 \times 99

.

For

a - b \geq 10

, we need

a \geq b + 10

.

For each value of

a

from 11 to 100, the number of valid b values is

a - 10

.

Favorable cases:

\sum_{a = 11}^{100} (a - 10) = \sum_{k = 1}^{90} k = \frac{90 \cdot 91}{2} = 4095

.

Probability:

\frac{4095}{9900} = \frac{91}{220}

(after simplification).

Since

g cd (91, 220) = 1

, we have

m = 91, n = 220

.

Thus

m + n = 311

JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Integer answer