There are nine points on a piece of paper. No three of the points are collinear. How many different triangles can be formed by using three of the nine points as vertices?
Number of possible choices for the first point = 9.
Number of possible choices for the second point = 8.
Number of possible choices for the third point = 7.
Possible triangles with specific order of points = 9*8*7 = 504.
However, when we say triangle, we are not really concerned in which order the points are selected. So we have over-counted the number of triangles by 6, which is the number of ways to order three points.
The number of distinct triangles is therefore 504/6=84.
This number is mathematically called
9 choose 3, calculated by
9!/(3!(9-3)!) = 84
where 9! is factorial 9, = 9*8*7*...*2*1
To find out the number of different triangles that can be formed, we can use the combination formula, also known as "n choose k."
In this case, we have 9 points, and we want to choose 3 points to form a triangle.
The formula for combinations is given by:
C(n, k) = n! / (k! * (n-k)!)
Where C(n, k) represents the number of ways to choose k items from a set of n distinct items.
Using this formula, we can calculate the number of triangles:
C(9, 3) = 9! / (3! * (9-3)!)
= 9! / (3! * 6!)
= (9 * 8 * 7) / (3 * 2 * 1)
= 84
Therefore, there are 84 different triangles that can be formed using three of the nine points as vertices.
To find the number of different triangles that can be formed using three of the nine points, we need to use the concept of combinations.
To form a triangle, we need to choose three points out of the nine available. The order in which we choose the points does not matter, as the same three points will always form the same triangle.
The number of ways to choose three points out of nine is given by the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Where n is the total number of items to choose from, and r is the number of items to choose.
In this case, we have n = 9 (the number of points) and r = 3 (the number of points to choose to form a triangle).
Plugging in the values, the formula becomes:
C(9, 3) = 9! / (3! * (9-3)!)
Simplifying,
C(9, 3) = (9 * 8 * 7) / (3 * 2 * 1)
C(9, 3) = 84
Therefore, there are 84 different triangles that can be formed using three of the nine points.