Difference between revisions of "1997 PMWC Problems/Problem I12"
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| − | 49 | + | == Problem == |
| + | In a die, 1 and 6, 2 and 5, 3 and 4 appear on opposite faces. When 2 dice are thrown, product of numbers appearing on the top and bottom faces of the 2 dice are formed as follows: | ||
| + | *number on top face of 1st die x number on top face of 2nd die | ||
| + | *number on top face of 1st die x number on bottom face of 2nd die | ||
| + | *number on bottom face of 1st die x number on top face of 2nd die | ||
| + | *number on bottom face of 1st die x number on bottom face of 2nd die | ||
| + | What is the sum of these 4 products ? | ||
| + | |||
| + | |||
| + | == Solution == | ||
| + | Let <math>x,y</math> be the two numbers at the top of the two dice. Then | ||
| + | |||
| + | <cmath>xy + x(7-y) + (7-x)y + (7-x)(7-y) = (x + (7-x))(y + (7 - y)) = 49</cmath> | ||
| + | |||
| + | == See also == | ||
| + | {{PMWC box|year=1997|num-b=I11|num-a=I13}} | ||
| + | |||
| + | [[Category:Introductory Combinatorics Problems]] | ||
Revision as of 19:07, 8 October 2007
Problem
In a die, 1 and 6, 2 and 5, 3 and 4 appear on opposite faces. When 2 dice are thrown, product of numbers appearing on the top and bottom faces of the 2 dice are formed as follows:
- number on top face of 1st die x number on top face of 2nd die
- number on top face of 1st die x number on bottom face of 2nd die
- number on bottom face of 1st die x number on top face of 2nd die
- number on bottom face of 1st die x number on bottom face of 2nd die
What is the sum of these 4 products ?
Solution
Let 
be the two numbers at the top of the two dice. Then
![\[xy + x(7-y) + (7-x)y + (7-x)(7-y) = (x + (7-x))(y + (7 - y)) = 49\]](http://latex.artofproblemsolving.com/e/6/3/e639f82969a61db66cf6cf03415f41313875ab5c.png)
See also
| 1997 PMWC (Problems) | ||
| Preceded by Problem I11 |
Followed by Problem I13 | |
| I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
