Finally, give the final number of triangles as an integer (i.e., no formulas) based on inclusion- exclusion and the quantities in 1,2 and 3. b. Now assume that the number of dots on one side of the arrangement is 12 and the number of dots on the other side of the shape is m, with one shared point at the bottom of the arrangement counting towards both n and m. (So, in our example diagram in part a., we have n = 5 and m = 4.) What is the total number of triangles that can be formed by choosing three points from the grid of size n and m? Generalize your solution from part a. and express your solution as a formula using n and m and inclusion-exclusion. Simplify your expression using the definition of the binomial coefficient so that it is expressed as a single fraction involving n and m. 2 Consider an organization of points as shown on the right. The points are organized into two lines that share an endpoint. We call this bottom-most point the "shared" point in this question. This question asks you to calculate how many triangles we can form from selecting three of these points. By triangle, we mean any selection of three end points from the points that results in a shape with three sides and an area that is bigger than zero. So, in particular, you can't pick three end points that are all in a line. Two examples of ways to choose the three points to create a triangle are illustrated here: In these images, the end points of the triangle are coloured in red. Notice that it does not matter that the lines of the triangle run through other points that are not endpoints. All that we care about is that we pick three points from the set of points and connect them to form a triangle. Any different choice of three valid (i.e., not in a line) end points will create a unique triangle. a. How many ways to make a triangle are there for the organization of points shown in the example? First, calculate these three quantities: 1. The number of triangles that choose two points on the left side and one point on the right side. (In this case, count the shared point as being one of the possible choices for the left- hand side, as in the example on the right.) 2. The number of triangles that choose one point on the left side and two points on the right side. (In this case, count the shared point as being one of the possible choices for the right- hand side.) 3. The number of triangles that use the shared point. Use a binomial coefficient to express each of 1 and 2. The binomial coefficient should use actual integers (3,4,5, etc.), not unknowns like n or k.
Finally, give the final number of triangles as an integer (i.e., no formulas) based on inclusion- exclusion and the quantities in 1,2 and 3. b. Now assume that the number of dots on one side of the arrangement is 12 and the number of dots on the other side of the shape is m, with one shared point at the bottom of the arrangement counting towards both n and m. (So, in our example diagram in part a., we have n = 5 and m = 4.) What is the total number of triangles that can be formed by choosing three points from the grid of size n and m? Generalize your solution from part a. and express your solution as a formula using n and m and inclusion-exclusion. Simplify your expression using the definition of the binomial coefficient so that it is expressed as a single fraction involving n and m. 2 Consider an organization of points as shown on the right. The points are organized into two lines that share an endpoint. We call this bottom-most point the "shared" point in this question. This question asks you to calculate how many triangles we can form from selecting three of these points. By triangle, we mean any selection of three end points from the points that results in a shape with three sides and an area that is bigger than zero. So, in particular, you can't pick three end points that are all in a line. Two examples of ways to choose the three points to create a triangle are illustrated here: In these images, the end points of the triangle are coloured in red. Notice that it does not matter that the lines of the triangle run through other points that are not endpoints. All that we care about is that we pick three points from the set of points and connect them to form a triangle. Any different choice of three valid (i.e., not in a line) end points will create a unique triangle. a. How many ways to make a triangle are there for the organization of points shown in the example? First, calculate these three quantities: 1. The number of triangles that choose two points on the left side and one point on the right side. (In this case, count the shared point as being one of the possible choices for the left- hand side, as in the example on the right.) 2. The number of triangles that choose one point on the left side and two points on the right side. (In this case, count the shared point as being one of the possible choices for the right- hand side.) 3. The number of triangles that use the shared point. Use a binomial coefficient to express each of 1 and 2. The binomial coefficient should use actual integers (3,4,5, etc.), not unknowns like n or k.
Chapter9: Math Models And Geometry
Section9.3: Use Properties Of Angles, Triangles, And The Pythagorean Theorem
Problem 108E: On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides the shown in the...
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Transcribed Image Text:Finally, give the final number of triangles as an integer (i.e., no formulas) based on inclusion-
exclusion and the quantities in 1,2 and 3.
b. Now assume that the number of dots on one side of the arrangement is 12 and the number of dots
on the other side of the shape is m, with one shared point at the bottom of the arrangement
counting towards both n and m. (So, in our example diagram in part a., we have n = 5 and m = 4.)
What is the total number of triangles that can be formed by choosing three points from the grid of
size n and m? Generalize your solution from part a. and express your solution as a formula using n
and m and inclusion-exclusion. Simplify your expression using the definition of the binomial
coefficient so that it is expressed as a single fraction involving n and m.
Transcribed Image Text:2
Consider an organization of points as shown on the right.
The points are organized into two lines that share an endpoint. We call
this bottom-most point the "shared" point in this question. This
question asks you to calculate how many triangles we can form from
selecting three of these points.
By triangle, we mean any selection of three end points from the points
that results in a shape with three sides and an area that is bigger than zero. So, in particular, you
can't pick three end points that are all in a line. Two examples of ways to choose the three points to
create a triangle are illustrated here:
In these images, the end points of the triangle are coloured in red. Notice that it does not matter
that the lines of the triangle run through other points that are not endpoints. All that we care about
is that we pick three points from the set of points and connect them to form a triangle. Any different
choice of three valid (i.e., not in a line) end points will create a unique triangle.
a. How many ways to make a triangle are there for the organization of points shown in the example?
First, calculate these three quantities:
1. The number of triangles that choose two points on the left side and one point on the right
side. (In this case, count the shared point as being one of the possible choices for the left-
hand side, as in the example on the right.)
2. The number of triangles that choose one point on the left side and two points on the right
side. (In this case, count the shared point as being one of the possible choices for the right-
hand side.)
3. The number of triangles that use the shared point.
Use a binomial coefficient to express each of 1 and 2. The binomial coefficient should use actual
integers (3,4,5, etc.), not unknowns like n or k.
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