2007 AMC 10B Problem 8

Question

On the trip home from the meeting where this AMC

10

was constructed, the Contest Chair noted that his airport parking receipt had digits of the form

bbcac,

where

0\le a\lt b\lt c\le 9,

and

b

was the average of

a

and

c.

How many different five-digit numbers satisfy all these properties?

12

16

18

20

24

The condition

b=\dfrac{a+c}{2}

forces

a

and

c

to have the same parity

Count pairs

a\lt c

among the even digits and among the odd digits; each pair fixes

b

Once

a

and

c

are chosen,

b=\dfrac{a+c}{2}

is determined, and

a\lt b\lt c

holds automatically. For

b

to be an integer,

a

and

c

must share parity.

Choosing two even digits from

\{0,2,4,6,8\}

gives

\binom{5}{2}=10

pairs, and choosing two odd digits from

\{1,3,5,7,9\}

gives another

10.

This yields

20

valid numbers.

Thus, the correct answer is D.

Problem 8 in other years