how many different tetrahedrons are determined by ______ given no four are coplanar?
4 points in space
9 points in space
15 points in space
n points in space
benis
To determine the number of different tetrahedrons, we can use the fact that no four points are coplanar. Let's analyze each case:
1. 4 points in space:
In this case, we have only 4 points. Since a tetrahedron requires 4 non-coplanar points, we can form only one tetrahedron.
2. 9 points in space:
To determine the number of tetrahedrons using 9 points, we need to select any 4 points that are not coplanar.
First, choose any 4 points from the 9 points available. This can be done in combination form, denoted as C(9, 4).
C(9, 4) = 9! / [(9-4)! * 4!] = 9! / (5! * 4!) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) = 126
Therefore, there are 126 different tetrahedrons that can be formed using 9 non-coplanar points.
3. 15 points in space:
Similar to the previous case, we need to choose any 4 points that are not coplanar. Using the combination formula, C(15, 4), we can calculate the number of tetrahedrons.
C(15, 4) = 15! / [(15-4)! * 4!] = 15! / (11! * 4!) = (15 * 14 * 13 * 12) / (4 * 3 * 2 * 1) = 1365
Therefore, there are 1365 different tetrahedrons that can be formed using 15 non-coplanar points.
4. n points in space:
By following the same pattern, for any given number of points 'n', you can calculate the number of tetrahedrons by using the combination formula C(n, 4).
C(n, 4) = n! / [(n-4)! * 4!]
Please note that if 'n' is less than 4, it is not possible to form a tetrahedron with fewer points.
By substituting the value of 'n' in the formula, you can determine the number of tetrahedrons for that specific case.
Remember, the condition that no four points are coplanar is essential for calculating the number of tetrahedrons.