Probability Part 3

Concept:The product $ab$ is divisible by $3$ if at least one of $a$ or $b$ is a multiple of $3$.We use complementary counting: find pairs where neither number is divisible by $3$, then subtract from total.Formula:Total ways to choose two distinct numbers from $n$: $\binom{n}{2} = \frac{n(n-1)}{2}$.Complementary probability: $P(\text{divisible}) = 1 - P(\text{not divisible})$.Solution:Total numbers from $1$ to $50$: $50$.Total ways to select two distinct numbers: $\binom{50}{2} = 1225$.Multiples of $3$ in $\{1,2,\ldots,50\}$: $\left\lfloor \frac{50}{3} \right\rfloor = 16$.Numbers not divisible by $3$: $50 - 16 = 34$.Unfavorable pairs (both numbers not divisible by $3$): $\binom{34}{2} = \frac{34 \cdot 33}{2} = 561$.Favorable pairs (product divisible by $3$): $1225 - 561 = 664$.Required probability: $\frac{664}{1225}$.Answer:Option C: $\frac{664}{1225}$.