Question
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 \_\_\_\_.
JEE Main
Mathematics
Probability
2026
JEE Main 2026 (23 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. A
- B. B
- C. C
- D. D
Answer
A. A
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 ^ 90 k = 90 91 2 = 4095. Probability = 4095 9900 = 91 220 (after dividing by 45). (91, 220) = 1, so m = 91, n = 220. m + n = 311.
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