7. Two cards are chosen, without replacement, from a deck of 50 cards numbered $1,2,3, \ldots, 50$. What is the probability that the product of the numbers on these cards is divisible by 7 ?
Answer: $\frac{46}{175}$
Solution:
Note that there are $\left\lfloor\frac{50}{7}\right\rfloor=7$ multiples of 7 from 1 to 50 , so there are 43 numbers which are not divisible by 7. The probability that after choosing two cards, the product of these numbers is not divisible by 7 is $\binom{43}{2} /\binom{50}{2}=\frac{43 \cdot 42}{50 \cdot 49}=\frac{129}{175}$. Hence, the probability in question is then $1-\frac{129}{175}=\frac{46}{175}$.