Question

The number of straight lines that can be drawn out of 10 points of which 7 are collinear is (a) 22 (b) 23 (c) 24 (d) 25

Short answer

The number of straight lines that can be drawn out of 10 points, of which 7 are collinear, is 25. We calculate this by considering two cases: (1) both points for the line are chosen from the 7 collinear points, and (2) at least one of the points is chosen from the 3 non-collinear points. After calculating the combinations for both cases, we should subtract the extra count of the straight line formed by the 7 collinear points. The correct answer is 25 lines, corresponding to option (d).

Step-by-step solution

Case 1: Both points are from the 7 collinear points

We have to choose 2 points from the 7 collinear points. The number of ways to do this can be calculated using combinations, which is denoted as C(n, r), where n is the total number of elements to choose from and r is the number of elements to choose. So in this case, it is C(7, 2): \[ C(7, 2) = \frac{7!}{2!(7-2)!} = \frac{7!}{2!5!} = \frac{7\times 6}{2} = 21 \] So, 21 lines can be drawn when both points are chosen from the 7 collinear points.

Case 2: At least one point chosen from the 3 non-collinear points

Now we have to calculate the number of lines formed when at least one point is from the 3 non-collinear points. To do this, we can choose 2 points from the remaining 10 - 7 = 3 non-collinear points: \[ C(3, 2) = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!} = \frac{3\times 2}{2} = 3 \] Next, we need to count the lines formed when one point is from the collinear points and the other is from the non-collinear points. There are 7 collinear points and 3 non-collinear points, so we have: \[C(7, 1) * C(3, 1) = (7 * 3) = 21\] Now, we add the number of lines formed by choosing two non-collinear points (3) to the number of lines formed by choosing one collinear and one non-collinear point (21): \[3 + 21 = 24\]

Final answer

We now add the lines formed in both cases to get the final answer: \[21 + 24 = 45\] However, in this solution, we have counted the straight line formed by the 7 collinear points multiple times. Since there is only one such line, we need to subtract the extra count, which is 21 lines minus 1 correct line: \[ 45 - (21 - 1) = 45 - 20 = 25\] Therefore, the correct answer is 25 lines, which corresponds to option (d).

Key concepts

Collinear Points

In geometry, collinear points are three or more points that lie on the same straight line. When we say that points are collinear, it simply means that they fall in a straight line without deviating. Understanding this is crucial because when dealing with collinear points, we recognize that any line segment drawn between any two points within this group still results in the same line, thus not forming a new distinct line.
In problems involving lines and geometry, acknowledging collinear points helps in accurately calculating the number of unique straight lines that can be formed from a set of points. In the context of our exercise, 7 out of the 10 points are collinear, meaning all these 7 points contribute to just one unique straight line by themselves.

Combination Formula

The combination formula is a fundamental concept in combinatorics, used to determine how many ways we can select a group of items from a larger group, where the order of selection does not matter. It is expressed mathematically as:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

Where:

• \( n \) is the total number of items,
• \( r \) is the number of items to choose,
• \(!\) is the factorial notation, meaning the product of all positive integers up to a given number.

In the exercise, the combination formula is used to calculate possible ways of selecting points from the 7 collinear points and also from the 3 non-collinear points. For instance, choosing 2 points out of the 7 collinear points uses this formula: \( C(7, 2) \). This calculation is pivotal to determine the number of potential line formations.

Non-collinear Points

Non-collinear points refer to points that do not all lie on a single straight line. When considering non-collinear points, each pair of points can give rise to a unique straight line. This concept is significant because it increases the diversity in the types of lines that can be formed when the points do not all align with each other.
In the given problem, 3 points are non-collinear among the total of 10 points. This affects calculations significantly; each selection of two points from these three results in a unique line. Using the combination formula as \( C(3, 2) \), we find that there are 3 lines formed by these non-collinear points. It is this variety introduced by non-collinear points that complements the otherwise single-line outcome of collinear points.

Straight Lines in Geometry

A straight line is a line with no curves and whose points extend in both directions infinitely. It is the simplest shape in geometry and forms the backbone of geometric principles and calculations. In problems like the exercise we're analyzing, it's essential to understand that a straight line can be formed between any two distinct points.
When determining the number of straight lines that can be formed from multiple points, considerations must include:

• Identifying any sets of collinear points, as they do not form additional lines.
• Using combinations to determine how many distinct pairs of points can be drawn from the set.
• Ackacting the uniqueness of lines formed by non-collinear points.

Incorporating these principles allows us to solve for the number of distinct lines accurately. Hence, for 10 points where 7 are collinear, understanding these concepts is critical to determine that ultimately 25 unique lines can be formed.