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Difference between revisions of "2017 AMC 10A Problems/Problem 23"

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Finally, we subtract all the ones in a line from <math>2300</math>, so we have <math>2300-120-16-4-12=\boxed{(\textbf{B})\ 2148}</math>
 
Finally, we subtract all the ones in a line from <math>2300</math>, so we have <math>2300-120-16-4-12=\boxed{(\textbf{B})\ 2148}</math>
  
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==Solution 2 (Pick's Theorem)==
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==Solution 2 (Concise)==
  
 
There are <math>5 \times 5 = 25</math> total points in all. So, there are <math>\dbinom{25}3 = 2300</math> ways to choose the three vertices for the triangle. However, there are some cases where they 3 points chosen results in a straight line.  
 
There are <math>5 \times 5 = 25</math> total points in all. So, there are <math>\dbinom{25}3 = 2300</math> ways to choose the three vertices for the triangle. However, there are some cases where they 3 points chosen results in a straight line.  
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There are <math>3 \times 4=12</math> ways where the 3 points make the a slope of <math>\frac{1}{2}</math>, <math>-\frac{1}{2}</math>, <math>2</math>, and <math>-2</math>.
 
There are <math>3 \times 4=12</math> ways where the 3 points make the a slope of <math>\frac{1}{2}</math>, <math>-\frac{1}{2}</math>, <math>2</math>, and <math>-2</math>.
  
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Hence, there are <math>100+40+12=152</math> cases where the chosen 3 points make a line. Every triangle in the grid will produce an area that is either a whole number or a number that ends in <math>.5</math>. Since Pick's Theorem states that the area is <math>\text{lattice points} + \frac{\text{boundary points}}{2} - 1</math>. If the number of boundary points is even, then the area is a whole number. If not, then it ends in <math>.5</math> (because any odd number divided by 2 ends in <math>.5</math>)
+
Hence, there are <math>100+40+12=152</math> cases where the chosen 3 points make a line. The answer would be <math>2300-152=\boxed{(\textbf{B})\ 2148}</math>
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Therefore, we can safely state that the answer is <math>2300-152=\boxed{(\textbf{B})\ 2148}</math>
 
  
 
~MrThinker
 
~MrThinker

Revision as of 21:20, 13 October 2022

Problem

How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?

$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$


Solution 1

We can solve this by finding all the combinations, then subtracting the ones that are on the same line. There are $25$ points in all, from $(1,1)$ to $(5,5)$, so $\dbinom{25}3$ is $\frac{25\cdot 24\cdot 23}{3\cdot 2 \cdot 1}$, which simplifies to $2300$. Now we count the ones that are on the same line. We see that any three points chosen from $(1,1)$ and $(1,5)$ would be on the same line, so $\dbinom53$ is $10$, and there are $5$ rows, $5$ columns, and $2$ long diagonals, so that results in $120$. We can also count the ones with $4$ on a diagonal. That is $\dbinom43$, which is 4, and there are $4$ of those diagonals, so that results in $16$. We can count the ones with only $3$ on a diagonal, and there are $4$ diagonals like that, so that results in $4$. We can also count the ones with a slope of $\frac12$, $2$, $-\frac12$, or $-2$, with $3$ points in each. Note that there are $3$ such lines, for each slope, present in the grid. In total, this results in $12$. Finally, we subtract all the ones in a line from $2300$, so we have $2300-120-16-4-12=\boxed{(\textbf{B})\ 2148}$

Solution 2 (Concise)

There are $5 \times 5 = 25$ total points in all. So, there are $\dbinom{25}3 = 2300$ ways to choose the three vertices for the triangle. However, there are some cases where they 3 points chosen results in a straight line.

There are $10 \times 10 = 100$ cases where the 3 points chosen make up a vertical or horizontal line.

There are $2\left(1+\dbinom{4}3+\dbinom{5}3+\dbinom{4}3+1\right)=40$ cases where the 3 points all land on the diagonals of the square.

There are $3 \times 4=12$ ways where the 3 points make the a slope of $\frac{1}{2}$, $-\frac{1}{2}$, $2$, and $-2$.

Hence, there are $100+40+12=152$ cases where the chosen 3 points make a line. The answer would be $2300-152=\boxed{(\textbf{B})\ 2148}$

~MrThinker

See Also

Video Solution:

https://www.youtube.com/watch?v=wfWsolGGfNY

https://www.youtube.com/watch?v=LCvDL-SMknI


2017 AMC 10A (Problems • Answer Key • Resources)
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Problem 24
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