1981 AHSME Problems/Problem 22
Problem 22
How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form 
, where 
, 
, and 
are positive integers not exceeding four?
Solution 1(casework)
Restating the problem, we seek all the lines that will pass through (, , ), (
, 
, 
), (
, 
, 
), and (
, 
, 
), such that 
are positive integers, 
are integers, and all of our points are between 1 and 4, inclusive. With this constraint in mind, we realize that for each coordinate, we have three choices:
- Set
to 
. This then allows us to set the corresponding 
to any number from 
to 
, inclusive. - Set to . This forces us to set the corresponding
to . - Set to
. This forces us to set the corresponding to .
to 
. This then allows us to set the corresponding 
to 
to 
. This forces us to set the corresponding Note that options 2 and 3 will give us the same points if we mirror the assignments of each coordinate. Also note that we cannot set all three coordinates to not change, as that would be a point.
All of this gives us 
ways to assign each coordinate, for a total of 
. We then must subtract the ways to get a point (
ways per coordinate, for a total of 
). This leaves us with 
. Finally, we divide by 
to account for mirror assignments giving us the same coordinate, for a final answer of 
.
(This was my first solution, apologies if it is bad).
See also
| 1981 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
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