Difference between revisions of "1981 AHSME Problems/Problem 22"
m (→Solution 1(casework)) |
(→Solution 1(casework)) |
||
Line 9: | Line 9: | ||
# Set <math>a/b/c</math> to <math>1</math>. This forces us to set the corresponding <math>i/j/k</math> to <math>1</math>. | # Set <math>a/b/c</math> to <math>1</math>. This forces us to set the corresponding <math>i/j/k</math> to <math>1</math>. | ||
# Set <math>a/b/c</math> to <math>-1</math>. This forces us to set the corresponding <math>i/j/k</math> to <math>4</math>. | # Set <math>a/b/c</math> to <math>-1</math>. This forces us to set the corresponding <math>i/j/k</math> to <math>4</math>. | ||
− | Note that options 2 and 3 will give us the same | + | 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. |
<br> | <br> | ||
All of this gives us <math>6</math> ways to assign each coordinate, for a total of <math>216</math>. We then must subtract the ways to get a point (<math>4</math> ways per coordinate, for a total of <math>64</math>). This leaves us with <math>152</math>. Finally, we divide by <math>2</math> to account for mirror assignments giving us the same coordinate, for a final answer of <math>76</math>. | All of this gives us <math>6</math> ways to assign each coordinate, for a total of <math>216</math>. We then must subtract the ways to get a point (<math>4</math> ways per coordinate, for a total of <math>64</math>). This leaves us with <math>152</math>. Finally, we divide by <math>2</math> to account for mirror assignments giving us the same coordinate, for a final answer of <math>76</math>. | ||
<br> | <br> | ||
(This was my first solution, apologies if it is bad). | (This was my first solution, apologies if it is bad). |
Latest revision as of 19:46, 9 September 2024
Problem
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
.
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).