1985 AJHSME Problems/Problem 22

Problem

Assume every 7-digit whole number is a possible telephone number except those that begin with $0$ or $1$ . What fraction of telephone numbers begin with $9$ and end with $0$ ?

$\text{(A)}\ \frac{1}{63} \qquad \text{(B)}\ \frac{1}{80} \qquad \text{(C)}\ \frac{1}{81} \qquad \text{(D)}\ \frac{1}{90} \qquad \text{(E)}\ \frac{1}{100}$

Note: All telephone numbers are 7-digit whole numbers.

Solution

An equivalent problem is finding the probability that a randomly selected telephone number begins with $9$ and ends with $0$ .

There are $10-2=8$ possibilities for the first digit in total, and only $1$ that works, so the probability the number begins with $9$ is $\frac{1}{8}$

There are $10$ possibilities for the last digit, and only $1$ that works $(0)$ , so the probability the number ends with $0$ is $\frac{1}{10}$

Since these are independent events, the probability both happens is $\frac{1}{8}\cdot \frac{1}{10}=\frac{1}{80}$

$\boxed{\text{B}}$

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1985 AJHSME Problems/Problem 22

Problem

Solution

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