Two distinct numbers a and b are selected at random from 1,2,3, , 50.

Question

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

Accepted Answer

Total ways to choose two distinct numbers from $1,2,\ldots ,50$ is

$(\frac{50}{2})=1225.$

The product $ab$ is not divisible by $3$ only if neither $a$ nor $b$ is divisible by $3$.

Multiples of $3$ in $1$ to $50$: $\lfloor \frac{50}{3}\rfloor =16$.

So numbers not divisible by $3$: $50-16=34$.

Unfavorable pairs:

$(\frac{34}{2})=\frac{34\cdot 33}{2}=561.$

Therefore favorable pairs:

$1225-561=664,$

and the required probability is

$\frac{664}{1225}.$

Answer: $\frac{664}{1225}$ (Option C).

Solutions · 1