Question

A three figure number is formed by the digits $4,5,6,7,8$ (no digit being repeated in any number). The probability that the number formed is divisible by $9$ is

• A. $\frac{1}{5}$
• B. $\frac{1}{10}$
• C. $\frac{1}{15}$
• D. $\frac{1}{6}$

Answer 1

Answer: (A)

$\frac{1}{5}$

A number is divisible by $9$ if its sum of digits is divisible by $9$. The digits used are: $4,5,6,7,8$. Out of these, only $3$ are to be used.
Sum of all the digits = $30$.
We need to remove $2$ numbers from this so that the total becomes a multiple of $9$, i.e. $27$ or $18$ or $9$.
We cannot reduce the total to $27$ by removing $2$ numbers. Neither can we reduce the total to $9$.
To make $18$, we need to remove $12$ i.e. $4\&8$ OR $5\&7$.
Thus, there are two possible cases:
(i) $5,6,7$
(ii) $4,6,8$
Thus, required probability = $\frac{2\ast 3!}{5\ast 4\ast 3}=\frac{1}{5}$
Hence, (a) is correct.