jmc

The fraction is simply the number of $7$-digit phone numbers with the restrictions applied divided by the total number of phone numbers. Let $a$ denote the numerator, and $b$ denote the denominator. Let's first work on finding $b$, the total number. For a regular digit, there are $10$ possible choices to make: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, or $9$. The only digit that is not regular is the first one, which prohibits $0$ and $1$ from taking place, resulting in $8$ possible choices to make for that first digit. Since each digit is independent of one another, we multiply the number of choices for each digit, resulting in $8\ast 10\ast 10\ast 10\ast 10\ast 10\ast 10$, or $8\ast 10^6$ possible total phone numbers ($b$). Now that we have the denominator, the only unknown remaining is $b$. To solve for $a$, let's use the same method as we did for the denominator. For the first digit, there is only one possible value: $9$. For the last digit, there is only one possible value: $0$. However, the rest of the five digits are "free" (meaning they can be any one of $10$ choices). Thus $a=1\ast 10\ast 10\ast 10\ast 10\ast 10\ast 1$, or $10^5$ possible phone numbers with restrictions. The fraction $\frac{a}{b}$ is the same as $\frac{10^5}{8\ast 10^6}$, which reduces to $\boxed{\frac{1}{80}}$. (~thelinguist46295)