1985 AJHSME Problems/Problem 22

From AoPSWiki

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 $0$ ?

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

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 $\frac{1}{8}\cdot \frac{1}{10}=\frac{1}{80}$

$\boxed{\text{B}}$

1985 AJHSME (Problems • Resources)
Preceded by Problem 21	Followed by Problem 23
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Personal tools

Navigation

Search

Toolbox

1985 AJHSME Problems/Problem 22

From AoPSWiki

Problem

Solution

See Also