Probability
Product Divisible by 3 — Complementary Counting
nta_pyq_2026_jan
Grade 12
Question:
Two distinct numbers $a$ and $b$ are selected at random from $1,2,3,\ldots,50$. The probability that their product $ab$ is divisible by 3, is
$\dfrac{272}{1225}$
$\dfrac{561}{1225}$
$\dfrac{664}{1225}$
$\dfrac{8}{25}$
Step-by-Step Solution
Key Concept: $ab$ divisible by 3 $\Leftrightarrow$ at least one of $a,b$ is divisible by 3. Multiples of 3 in $\{1,\ldots,50\}$: 16. Non-multiples: 34. $P(\text{neither divisible by }3)=\tfrac{\binom{34}{2}}{\binom{50}{2}}=\tfrac{561}{1225}$.
$P=\dfrac{664}{1225}$.
Correct Answer: 3