Question

A plane contains 12 points of which 4 are collinear. How many different straight lines can be formed with these points? (1) 50 (2) 66 (3) 60 (4) 61

Short answer

Answer: 29 different straight lines can be formed with these points.

Step-by-step solution

Find the number of lines that can be formed with non-collinear points.

There are a total of 12 points, and we know that 4 of them are collinear. This means that there are 12 - 4 = 8 non-collinear points. To find the number of lines that can be formed using these 8 non-collinear points, we can use the combination formula which is given by: C(n, k) = n! / (k!(n-k)!) For our case, n = 8 (number of non-collinear points) and k = 2 (as 2 points can form a straight line). C(8, 2) = 8! / (2!(8-2)!) = 8! / (2!6!) = 28 So, there are 28 different lines that can be formed using the 8 non-collinear points.

Find the number of lines that can be formed with the collinear points.

There are 4 collinear points and since they are on the same line, there is only 1 line that can be formed using these 4 points.

Calculate the total number of lines.

To find the total number of lines, we'll simply add the number of lines formed by non-collinear points and the number of lines formed by the collinear points. Total lines = 28 (from non-collinear points) + 1 (from collinear points) = 29 Therefore, our answer is not among the given options. The correct answer should be 29 different straight lines that can be formed with these points.

Key concepts

Combination Formula

Understanding the combination formula is essential in combinatorics, a branch of mathematics concerned with counting, combination, and permutation of sets. The formula is used when the order of selection does not matter and is represented as C(n, k) = \( \frac{n!}{k!(n-k)!} \), where:

• \( n \) denotes the total number of items,
• \( k \) represents the number of items to be selected,
• \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).

In our exercise, we calculated the number of straight lines that can be formed from 8 non-collinear points, choosing 2 at a time. This reflects real-life situations where certain combinations of elements are needed without concern for the sequence in which they are chosen.
To improve understanding, picture a scenario where you're choosing two students from a group for a team. The team's composition is what matters, not who was chosen first or second; this is where the combination formula applies.

Straight Lines Determination

The concept of determining the number of straight lines is crucial in geometry. A straight line in a plane is uniquely determined by two distinct points. Therefore, given a set of points, if we wish to calculate how many unique straight lines can emerge, we would look at all possible pairs of these points.
However, there is a caveat: one must ensure that no more than two points are collinear, meaning lying on the same straight line. In our exercise, the presence of collinear points influenced our calculation, because collinear points offer fewer lines than expected. A robust understanding of this nuance can deepen students' grasp of geometric properties and spatial relationships.

Collinear Points

Visualizing Collinear Points

Imagine three or more points lying exactly on the same straight line; these are referred to as collinear points. In real-world terms, think of beads on a string that are perfectly aligned; this alignment signifies collinearity. In our exercise, we determined that the 4 collinear points only contribute one line to the total count, despite there being multiple points. This is because no matter which two points you choose out of these collinear ones, they will always form the same straight line. Hence, when counting lines in combinatorics, collinear points play a unique role, and understanding this helps in avoiding overcounting.

Non-Collinear Points

Understanding Non-Collinear Points

On the flip side, non-collinear points do not all lie on the same straight line. When given several points, if you select any three and they do not line up perfectly, these points are non-collinear. In geometric exercises, non-collinear points are exciting because they increase the potential for creating more straight lines; each pair generates a unique line. The exercise at hand involved 8 non-collinear points, essential in calculating the number of lines since collinear ones could not produce additional unique lines. Recognizing and applying the concept of non-collinearity ensures a proper count and encourages a deeper understanding of geometric principles in applied contexts.