Difference between revisions of "2008 UNCO Math Contest II Problems/Problem 1"
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− | == Solution == | + | == Solution == |
+ | Case 1: The numbers include <math>0,1,2</math> | ||
+ | Row 1: <math>0 1 1</math> | ||
+ | |||
+ | Row 2: <math>2 0 0</math> | ||
+ | |||
+ | Row 3: <math>0 1 1</math> | ||
+ | |||
+ | There are 3 rows to choose where to place Row 2, giving us <math>\boxed3</math> cases. | ||
+ | |||
+ | Case 2: The numbers include <math>0,1,2</math> | ||
+ | |||
+ | Row 1: <math>1 0 1</math> | ||
+ | |||
+ | Row 2: <math>0 2 0</math> | ||
+ | |||
+ | Row 3: <math>1 0 1</math> | ||
+ | |||
+ | There are 3 rows to choose where to place Row 2, giving us <math>\boxed3</math> cases. | ||
+ | |||
+ | Case 3: The numbers include <math>0, 1, 2</math> | ||
+ | |||
+ | Row 1: <math>1 1 0</math> | ||
+ | |||
+ | Row 2: <math>0 0 2</math> | ||
+ | |||
+ | Row 3: <math>1 1 0</math> | ||
+ | |||
+ | There are 3 rows to choose where to place Row 2, giving us <math>\boxed3</math> cases. | ||
+ | |||
+ | Case 4: The numbers include <math>0, 2</math> | ||
+ | |||
+ | Row 1: <math>0 0 2</math> | ||
+ | |||
+ | Row 2: <math>0 2 0</math> | ||
+ | |||
+ | Row 3: <math>2 0 0</math> | ||
+ | |||
+ | There are 3 rows to choose where to place Row 2, and Row 1 can go on either top or bottom, giving us <math>3\times2=\boxed6</math> cases. | ||
+ | |||
+ | Case 5: The numbers include <math>0, 1</math> | ||
+ | |||
+ | Row 1: <math>0 1 1</math> | ||
+ | |||
+ | Row 2: <math>1 0 1</math> | ||
+ | |||
+ | Row 3: <math>1 1 0</math> | ||
+ | |||
+ | There are 3 rows to choose where to place Row 2, and Row 1 can go on either top or bottom, giving us <math>3\times2=\boxed6</math> cases. | ||
+ | |||
+ | This gives us a total of <math>3+3+3+6+6=\boxed{21}</math> ways. | ||
== See Also == | == See Also == |
Latest revision as of 12:53, 11 January 2019
Problem
Determine the number of square arrays whose row and column sums are equal to , using as entries. Entries may be repeated, and not all of need be used as the two examples show.
Solution
Case 1: The numbers include
Row 1:
Row 2:
Row 3:
There are 3 rows to choose where to place Row 2, giving us cases.
Case 2: The numbers include
Row 1:
Row 2:
Row 3:
There are 3 rows to choose where to place Row 2, giving us cases.
Case 3: The numbers include
Row 1:
Row 2:
Row 3:
There are 3 rows to choose where to place Row 2, giving us cases.
Case 4: The numbers include
Row 1:
Row 2:
Row 3:
There are 3 rows to choose where to place Row 2, and Row 1 can go on either top or bottom, giving us cases.
Case 5: The numbers include
Row 1:
Row 2:
Row 3:
There are 3 rows to choose where to place Row 2, and Row 1 can go on either top or bottom, giving us cases.
This gives us a total of ways.
See Also
2008 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 |