JEE Main · MathematicsHard

$Two numbers are selected at random from the set {1, 2, \dots, 100} . What is the probability that their sum is divisible by 3 ?$

A. $33/99$
B. $11/33$
C. $32/99$
D. $1/3$

Show correct answer & step-by-step solution

Correct answer: A — $33/99$

Solution

Let S_{0}, S_{1}, S_{2} be subsets with remainders 0, 1, 2. ∣ S_{0} ∣ = 33, ∣ S_{1} ∣ = 34, ∣ S_{2} ∣ = 33 . Sum is divisible by 3 if both are from S_{0} or one from S_{1} and one from S_{2} . Total pairs = (2 100) = 4950 . Favorable = (2 33) + 34 \times 33 = 528 + 1122 = 1650 . Probability = $1650/4950 = 1/3 = 33/99$.

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