2009 UNCO Math Contest II Problems/Problem 1
Problem
How many positive 
-digit numbers 
are there such that 
For example, 
and 
have this property but 
and 
do not.
Solution
Here, it is a slight trick.
Here, we need to find 
such that 
, 
and where 
.
If we place 
, then we can place 
as 
, i.e. in 
ways.
Similarly, if we place 
, we can place 
i.e. in 
ways.
![\[\dots\]](http://latex.artofproblemsolving.com/1/6/7/16762e47f22f53e2431935f211fa82570b46b6cb.png)
If, we place 
, we have the only choice 
, in 
ways.
So, in order to get the number of possibilities, we have to add the no. of all the possibilities we got, i.e. the answer is ![\[\color{red}{1+2+3+4+5+6+7+8+9=\frac {9\times 10}{2}}=\color{blue}{45}\]](http://latex.artofproblemsolving.com/d/3/f/d3f93e37c8802607aedddfdcb9905f32e8f74b00.png)
See also
| 2009 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
