Difference between revisions of "2013 AIME II Problems/Problem 11"
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Let <math>A = \{1, 2, 3, 4, 5, 6, 7\}</math>, and let <math>N</math> be the number of functions <math>f</math> from set <math>A</math> to set <math>A</math> such that <math>f(f(x))</math> is a constant function. Find the remainder when <math>N</math> is divided by <math>1000</math>. | Let <math>A = \{1, 2, 3, 4, 5, 6, 7\}</math>, and let <math>N</math> be the number of functions <math>f</math> from set <math>A</math> to set <math>A</math> such that <math>f(f(x))</math> is a constant function. Find the remainder when <math>N</math> is divided by <math>1000</math>. | ||
| − | ==Solution | + | ==Solution 1== |
Any such function can be constructed by distributing the elements of <math>A</math> on three tiers. | Any such function can be constructed by distributing the elements of <math>A</math> on three tiers. | ||
Revision as of 18:08, 4 June 2020
Problem 11
Let 
, and let 
be the number of functions 
from set 
to set such that 
is a constant function. Find the remainder when is divided by 
.
Solution 1
Any such function can be constructed by distributing the elements of on three tiers.
The bottom tier contains the constant value, 
for any 
. (Obviously 
.)
The middle tier contains 
elements 
such that 
, where 
.
The top tier contains 
elements such that 
equals an element on the middle tier.
There are 
choices for 
. Then for a given , there are 
ways to choose the elements on the middle tier, and then 
ways to draw arrows down from elements on the top tier to elements on the middle tier.
Thus 
, giving the answer 
.
See Also
| 2013 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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. 
