1. Total number of five-digit numbers

The sample space $S$ consists of all five-digit numbers, i.e. from $10000$ to $99999$.

Hence,

2. Count five-digit multiples of 7

We need numbers in $[10000,99999]$ divisible by $7$.

• Smallest multiple of $7$ greater than or equal to $10000$:

so the first multiple is

• Largest multiple of $7$ less than or equal to $99999$:

so the last multiple is

Therefore, the number of five-digit multiples of $7$ is

3. Count five-digit numbers divisible by both 7 and 5

A number divisible by both $7$ and $5$ must be divisible by

So we count five-digit multiples of $35$.

• Smallest multiple of $35$ at least $10000$:

thus first such multiple is

• Largest multiple of $35$ at most $99999$:

thus last such multiple is

Hence, the number of five-digit multiples of $35$ is

4. Count numbers divisible by 7 but not by 5

Required count:

5. Compute the probability $p$

6. Compute $9p$

7. Match with the options

which matches Option C.