12 points lie on a circumference of a circle how many inscribed triangles can be constructed having 3 vertices?
To determine the number of inscribed triangles that can be constructed using 12 points lying on the circumference of a circle, we can use the combination formula.
The number of ways to choose 3 points out of 12 is calculated using the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Where:
n is the total number of points (12 in this case)
r is the number of points we want to choose for each triangle (3 in this case)
"!" denotes the factorial operation
Using the combination formula, we can calculate the number of ways to choose 3 points out of 12:
C(12, 3) = 12! / (3! * (12-3)! )
= 12! / (3! * 9! )
= 12 * 11 * 10 / (3 * 2 * 1)
= 220
Therefore, there are 220 different inscribed triangles that can be constructed using 12 points lying on the circumference of a circle.
To find the number of inscribed triangles that can be constructed using 12 points lying on the circumference of a circle, we can use the combination formula.
First, let's understand the concept. In an inscribed triangle, the vertices of the triangle must lie on the circumference of the circle.
To construct a triangle, we need 3 distinct points. The order of selecting the points doesn't matter in this case. Since we have 12 points on the circumference, we can select 3 points out of the 12.
The number of ways to select r objects out of a set of n objects is given by the combination formula:
C(n, r) = n! / (r! * (n-r)!)
where n! means factorial of n, which is the product of all positive integers from 1 to n.
In this case, n is 12 (the number of points) and r is 3 (the number of vertices in a triangle).
Therefore, the number of inscribed triangles that can be constructed is:
C(12, 3) = 12! / (3! * (12-3)! )
Simplifying the expression:
C(12, 3) = 12! / (3! * 9!)
The factorial of 12 is 12 * 11 * 10 * 9!
Cancelling out the common factor of 9!:
C(12, 3) = (12 * 11 * 10 * 9!) / (3! * 9!)
The 9! terms in the numerator and denominator cancel out:
C(12, 3) = (12 * 11 * 10) / (3 * 2 * 1)
Simplifying further:
C(12, 3) = 220
Therefore, there are 220 different triangles that can be constructed from the given 12 points lying on the circumference of a circle.