If three distinct numbers are chosen randomly from the first $100$ natural numbers, what is the probability that all three are divisible by both $2$ and $3$?

Explanation

Numbers divisible by both $2$ and $3$ are divisible by $6$ From $1$ to $100$, there are $16$ such numbers Number of ways to choose $3$ $= {}^{16}C_3$ Total possible selections $= {}^{100}C_3$ Probability $= {}^{16}C_3 / {}^{100}C_3 = 4/1155$