2022 AMC 8 Problems/Problem 23
Contents
Problem
A or
is placed in each of the nine squares in a
-by-
grid. Shown below is a sample configuration with three
s in a line.
How many configurations will have three
s in a line and three
s in a line?
Solution 1 (Casework)
s, one line of circles, and the last one can be anything that includes both shapes. There are ways to arrange the lines and
ways to choose the last lineththtrtrh
. In total, this is
Finally, we add and multiply: .
~wamofan
Solution 2
We will only consider columns, but at the end our answer should be multiplied by . There are
ways to choose a column for
and
ways to choose a column for
. The third column can be filled in
ways. Therefore, we have
ways. However, we overcounted the cases with
complete columns of with one symbol and
complete column with another symbol. This happens in
ways.
. However, we have to remember to double our answer giving us
.
~MathFun1000
Video Solution
https://www.youtube.com/watch?v=or4pKVzQ3gI
~Mathematical Dexterity
Video Solution
https://youtu.be/Ij9pAy6tQSg?t=2250
~Interstigation
Video Solution
https://www.youtube.com/watch?v=KYglbGTvfsY
Video Solution
https://youtu.be/0orAAUaLIO0?t=257
~STEMbreezy
Video Solution
~savannahsolver
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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