Question

Seven digits from the numbers $1,2,3,4,5,6,7,8$ and $9$ are written in random order. The probability that this seven-digit number is divisible by $9$ is $\frac{k}{9}$. Find $k$?

Answer 1

Total number of seven digit number is ${}^9{C}_7{=}^9{C}_2=36$
For divisibility with $9$, leave one of $(1,8),(2,7),(3,6),(4,5)$
Therefore required probability $=\frac{4}{36}=\frac{1}{9}$
$\therefore k=1$