JEE Main 2026 — Jan 22, Shift 1 — Mathematics solution

JEE Main2026Jan 22, Shift 1Mathematics

Two distinct numbers a and b are selected at random from 1, 2, 3, …, 50. The probability, that their product ab is divisible by 3, is

Solution

Use the complement: $P (ab divisible by 3) = 1 - P (neither a nor b divisible by 3)$ .

Multiples of 3 in ${1, \dots, 50}$ : $⌊ 50/3 ⌋ = 16$ . Non-multiples: $50 - 16 = 34$ .

Two distinct non-multiples chosen: $(2 34)$ ways out of $(2 50)$ total.

$P = 1 - \frac{( 2 34 )}{( 2 50 )} = 1 - \frac{34 \cdot 33}{50 \cdot 49} = 1 - \frac{1122}{2450} = 1 - \frac{561}{1225} = \frac{664}{1225}$ .

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