There are ten points in a plane, out of which four are in a straight l...
To solve this problem, we need to consider the different ways in which we can select four points from the given ten points to form a quadrilateral. Let's break down the solution into several steps:
Step 1: Identify the number of ways to select four points from the given ten points
To form a quadrilateral, we need to select four points from the ten given points. This can be done using combinations. The number of ways to select four points from ten can be calculated using the formula for combinations:
C(n, r) = n! / (r! * (n-r)!)
Here, n represents the total number of points and r represents the number of points to be selected.
Using this formula, we get:
C(10, 4) = 10! / (4! * (10-4)!)
= 10! / (4! * 6!)
= (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
= 210
So, there are 210 different ways to select four points from the given ten points.
Step 2: Subtract the cases where all four points lie on a straight line
Out of the 210 possible combinations, we need to subtract the cases where all four points lie on a straight line. Given that four points are collinear, we can choose any four points from the given four collinear points. This can be done using combinations as well:
C(4, 4) = 4! / (4! * (4-4)!)
= 4! / (4! * 0!)
= 1
So, there is only 1 way to select four points from the given four collinear points.
Step 3: Calculate the final number of quadrilaterals
To obtain the number of quadrilaterals formed by the given ten points, we subtract the cases where all four points are collinear from the total number of combinations:
Number of quadrilaterals = Total combinations - Collinear cases
= 210 - 1
= 209
Therefore, the correct answer is 209 quadrilaterals.
Since none of the provided options match the correct answer of 209, it seems that the options given in the question are incorrect or incomplete. The correct answer should be 209, not 185.
There are ten points in a plane, out of which four are in a straight l...
Make cases (4C0.6C4+ 4C1.6C3+ 4C2.6C2 )