Calculating Quadrilaterals From Points on a Plane

This problem requires us to determine the number of distinct quadrilaterals that can be formed by selecting vertices from a given set of points. We have 8 points in total on a plane. A key piece of information is that 4 of these points are collinear, meaning they lie on the same straight line. The remaining 4 points are not on this line.

A quadrilateral is a polygon with four sides and four vertices. To form a quadrilateral, we must select 4 points from the available points. Importantly, for a set of 4 points to form a valid quadrilateral, no three of those points can be collinear (lie on the same straight line).

Geometry Combinations Calculation

We can solve this problem using the principles of combinations. The formula for combinations, denoted as \(C(n, k)\) or \(\binom{n}{k}\), calculates the number of ways to choose \(k\) items from a set of \(n\) distinct items, where the order of selection does not matter. The formula is: \(C(n, k) = \frac{n!}{k!(n-k)!}\) where '\(!\)' denotes the factorial.

Step 1: Total Possible Selections of 4 Points

First, we calculate the total number of ways to choose any 4 points from the total 8 points available. This gives us all possible combinations of 4 points, irrespective of whether they form a quadrilateral.

Using the combination formula with \(n=8\) (total points) and \(k=4\) (points needed for a quadrilateral): \(C(8, 4) = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!}\) \(C(8, 4) = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1680}{24} = 70\)

So, there are 70 different ways to select a group of 4 points from the 8 points.

Step 2: Identifying Invalid Selections (Non-Quadrilaterals)

Next, we must identify and count the selections of 4 points that *do not* form a quadrilateral. This occurs when 3 or all 4 of the selected points lie on the same line.

Given that we have exactly 4 collinear points, the selections that fail to form quadrilaterals are:

• Case A: All 4 selected points are chosen from the group of 4 collinear points.

  The number of ways to choose 4 points from these 4 collinear points is:

  \(C(4, 4) = \frac{4!}{4!(4-4)!} = \frac{4!}{4!0!} = 1\)

  This single combination consists of points all lying on one line, so it cannot form a quadrilateral.

• Case B: Exactly 3 selected points are from the collinear group, and 1 point is chosen from the remaining 4 non-collinear points.

  To count these selections, we find the number of ways to choose 3 points from the 4 collinear points AND the number of ways to choose 1 point from the 4 non-collinear points.

  Ways to choose 3 collinear points: \(C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = 4\)

  Ways to choose 1 non-collinear point: \(C(4, 1) = \frac{4!}{1!(4-1)!} = \frac{4!}{1!3!} = 4\)

  Total ways for Case B = \(C(4, 3) \times C(4, 1) = 4 \times 4 = 16\)

  These 16 combinations include 3 collinear points, meaning they cannot form the vertices of a quadrilateral.

Step 3: Calculating Total Invalid Combinations

The total number of invalid selections is the sum of the counts from Case A and Case B.

Total Invalid Selections = (Ways from Case A) + (Ways from Case B)

Total Invalid Selections = \(1 + 16 = 17\)

Step 4: Calculating the Number of Valid Quadrilaterals

Finally, to find the number of actual quadrilaterals, we subtract the total count of invalid selections (from Step 3) from the total number of possible 4-point selections (from Step 1).

Number of Quadrilaterals = (Total Possible Selections) - (Total Invalid Selections)

Number of Quadrilaterals = \(70 - 17\)

\(\text{Number of Quadrilaterals} = 53\)

Thus, there are 53 distinct quadrilaterals that can be formed by joining the given 8 points, considering that 4 of them are collinear.