Question

All of David's telephone numbers have the form $555\text{–}abc\text{–}defg,$ where $a, b, c, d, e, f,$ and $g$ are distinct digits and in increasing order, and none is either $0$ or $1.$ How many different telephone numbers can David have?

$1$

$2$

$7$

$8$

$9$

Answer: D

Concepts:combinations bijection

Difficulty rating: 1510

Solution:

The seven digits are chosen from $\{2, 3, 4, 5, 6, 7, 8, 9\},$ and once chosen they must be written in increasing order, so only the choice of digits matters.

Choosing seven of these eight digits is the same as choosing the one digit to leave out, which can be done in $8$ ways.

Thus, D is the correct answer.

2005 AMC 10B Problem 18

Answer: D

Difficulty rating: 1510

Solution:

Problem 18 in Other Years