Difference between revisions of "1998 AHSME Problems/Problem 24"
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Revision as of 13:30, 5 July 2013
Problem
Call a 
-digit telephone number 
memorable if the prefix sequence 
is exactly the same as either of the sequences 
or 
(possibly both). Assuming that each 
can be any of the ten decimal digits 
, the number of different memorable telephone numbers is
Solution
Let 
represent the set of telephone numbers with 
(of which there are 
possibilities for 
and 
for 
), and 
those such that 
. Then 
(the telephone numbers that belong to both and ) is the set of telephone numbers such that 
, of which there are possibilities. By the Principle of Inclusion-Exclusion,
![\[|A \cup B| = |A| + |B| - |A \cap B| = 1000 \times 10 + 1000 \times 10 - 10 = 19990 \Rightarrow \mathrm{(C)}\]](http://latex.artofproblemsolving.com/3/f/e/3febd99dc1853249b3c76b1f8546190e0d6a2b5f.png)
See also
| 1998 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
