Difference between revisions of "2001 AIME I Problems/Problem 14"
Mathcool2009 (talk | contribs) |
Mathgeek2006 (talk | contribs) m (→Solution) |
||
Line 19: | Line 19: | ||
Clearly, <math>a_2=b_2=c_2=1</math>. Using the [[recursion|recursive]] equations and initial values: | Clearly, <math>a_2=b_2=c_2=1</math>. Using the [[recursion|recursive]] equations and initial values: | ||
− | <cmath>\begin{ | + | <cmath>\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} |
\multicolumn{19}{c}{}\\\hline | \multicolumn{19}{c}{}\\\hline | ||
n&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19\\\hline | n&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19\\\hline |
Revision as of 19:20, 10 March 2015
Problem
A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?
Solution
Let represent a house that does not receive mail and
represent a house that does receive mail. This problem is now asking for the number of
-digit strings of
's and
's such that there are no two consecutive
's and no three consecutive
's.
The last two digits of any -digit string can't be
, so the only possibilities are
,
, and
.
Let be the number of
-digit strings ending in
,
be the number of
-digit strings ending in
, and
be the number of
-digit strings ending in
.
If an -digit string ends in
, then the previous digit must be a
, and the last two digits of the
digits substring will be
. So
If an -digit string ends in
, then the previous digit can be either a
or a
, and the last two digits of the
digits substring can be either
or
. So
If an -digit string ends in
, then the previous digit must be a
, and the last two digits of the
digits substring will be
. So
Clearly, . Using the recursive equations and initial values:
\[\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \multicolumn{19}{c}{}\\\hline n&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19\\\hline a_n&1&1&1&2&2&3&4&5&7&9&12&16&21&28&37&49&65&86\\\hline b_n&1&2&2&3&4&5&7&9&12&16&21&28&37&49&65&86&114&151\\\hline c_n&1&1&2&2&3&4&5&7&9&12&16&21&28&37&49&65&86&114\\\hline \end{tabular}\] (Error compiling LaTeX. Unknown error_msg)
Therefore, the number of -digit strings is
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.