How many triangles are there with vertices
from the points shown below?
Now we are not allowing a degenerate triangles,
meaning ones with all vertices on the same line,
but we do allow non-right triangles.
For example, notice how if we select, let's say, this point,
this point, and this point, we can form a triangle.
It's just not a right triangle.
Also, notice, to form a right triangle,
one of the points selected would have to be
this corner point here.
So if this point is selected,
let's say this point, and this point,
notice how we do have a right triangle.
There is more than one way to determine
the number of possible triangles.
For this explanation, we will divide the points
into three groups, where one group contains just one point,
this corner point, and then we have this vertical group
of four points, and this horizontal group of five points.
To find all of the triangles, we will first determine
the number of non-right triangles that we can form
when we choose one point from the vertical group
and choose two points from the horizontal group.
And then we'll also determine the number
of non-right triangles we can form
when we choose two points from the horizontal group
and one point from the horizontal group.
That sum will give us all of the non-right triangles.
And then to determine the number of right triangles,
we need to choose one point from the vertical group,
one point from this corner group,
and one point from this horizontal group.
And once we add in the right triangles,
we will have all the possible triangles.
So, again, going back,
let's determine all the non-right triangles
when we choose one point from the vertical group
and two points from the horizontal group.
The order in which we select the points
does not affect the triangle,
and we don't use any point more than once,
and, therefore, we'll be using combinations.
To begin, the number of ways we can choose
one point from four is going to be 4
or, if we want, 4 choose 1,
and then times the number of ways we can choose
two points from five, which is 5 choose 2.
4 choose 1 X 5 choose 2 gives us the number
of non-right triangles formed when we choose
one point from the vertical group
and two points from the horizontal group.
And then we'll add in the number of non-right triangles
we can form when we choose
two points from the vertical group
and one point from the horizontal group.
Well, the number of ways of choosing
one point from five is just 5 or, if we want, 5 choose 1,
and then times the number of ways we can choose
two points from four is 4 choose 2.
So 5 choose 1 X 4 choose 2 gives us
the number of non-right triangles formed
when choosing two points from the vertical group
and one point from the horizontal group.
So this sum here gives us all of the non-right triangles,
and now we need to add in the number of right triangles.
So to form the right triangles, we would select one point
from each of the three groups.
The number of ways of choosing one point
from the vertical group of four points is 4 choose 1,
or just 4, and then times the number of ways of choosing
one point from one, which of course is just 1,
but I'll go ahead and list it as a combination,
and then times the number of ways of choosing
one point from five, which is 5 choose 1,
which of course is just 5, but I'll list it as 5 choose 1.
And now let's go and evaluate the combinations.
4 choose 1 is equal to 4,
5 choose 2 is equal to 10,
plus 5 choose 1 is equal to 5 X 4 choose 2 is 6,
plus 4 choose 1 is 4,
1 choose 1 is 1, and 5 choose 1 is 5.
This gives us 40 + 30 + 20, which is equal to 90 triangles.
I hope you found this helpful.