1998 AMC 12 Problem 24

Question

Call a

7

-digit telephone number

d_1d_2d_3

-

d_4d_5d_6d_7

memorable if the prefix sequence

d_1d_2d_3

is exactly the same as either of the sequences

d_4d_5d_6

or

d_5d_6d_7

(possibly both). Assuming that each

d_i

can be any of the ten decimal digits

0,

1,

2,

\ldots,

9,

the number of different memorable telephone numbers is

19{,}810

19{,}910

19{,}990

20{,}000

20{,}100

Count numbers satisfying each of the two matching conditions separately

If both matches hold, all seven digits are forced by one repeated digit

Each matching condition gives

10^4=10,000

numbers: choose the three prefix digits and the one unconstrained remaining digit. If both hold, then

d_1d_2d_3=d_4d_5d_6=d_5d_6d_7,

forcing all digits equal, so there are

10

overlaps. Inclusion-exclusion gives

10,000+10,000-10=19,990,

so C is correct.

Problem 24 in other years