2005 AMC 10B Problem 18

Problem 18 of 25IntermediateCounting & Probability

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

1.

How many different telephone numbers can David have?

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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.

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