Question

Seven digits from the numbers $1,2,3,4,5,6,7,8,9$ are written down non-repeatedly in a random order to form a seven-digit number. The probability that this seven-digit number is divisible by$9$is.

• A. $\frac{2}{9}$
• B. $\frac{7}{36}$
• C. $\frac{1}{9}$
• D. $\frac{7}{12}$

Answer 1

The correct option is C

$\frac{1}{9}$

Step 1: Find the number of elements in sample space

We know that the integer is divisible by $9$ if the sum of its digits is divisible by $9$.

Sum of the first $9$ natural numbers is

$n\left(S\right)=\frac{9\left(9+1\right)}{2}$

$n\left(S\right)=45$

$45$ is divisible by $9$.

So it must be the case that the sum of the two integers we don’t pick is divisible by $9$ to form the seven-digit number.

Number of ways of choosing two integers from $9$ integers is ${}^9{C}_2=36$

Step 2: Find the required probability

There are $4$ two-digit pairs whose sum is divisible by $9$.

$\left[(1,8),(2,7),(3,6),(4,5)\right]$

Take the ratio to get the probability that the seven-digit number so formed is divisible by $9$.

$$\begin{gathered} =\frac{4}{36} \\ =\frac{1}{9} \end{gathered}$$

Hence, the probability that this seven-digit number is divisible by$9$is $\frac{1}{9}$.

Hence, the correct option is C.