Question

There are 15 points in a plane. No three points are collinear except 5 points. How many different straight lines can be formed? (1) 105 (2) 95 (3) 96 (4) 106

Short answer

Answer: 96

Step-by-step solution

Calculate the total number of lines without restrictions

First, let's find out how many lines can be formed by selecting any two points out of the given 15 points. This can be done by using the combination formula which defines the number of ways to choose 'r' elements out of 'n' distinct elements as: C(n, r) = n! / (r! * (n-r)!) In this case, we need to choose 2 points out of 15, so the equation becomes: C(15, 2) = 15! / (2! * (15-2)!) C(15, 2) = 105

Subtract lines formed by collinear points

Now, as given in the problem, no three points are collinear except five points. So, out of the 105 lines calculated in step 1, we need to remove the lines formed by those collinear points. To do this, we first find the number of combinations of lines formed by those 5 collinear points: C(5, 2) = 5! / (2! * (5-2)!) C(5, 2) = 10

Add lines formed by collinear points as one line

Since the 5 collinear points form a single straight line, they contribute only one line. Thus, we need to add that one line back.

Calculate the total number of lines

Finally, to calculate the total number of different straight lines, subtract the lines formed by collinear points from the lines found without restrictions, and then add the one line that these collinear points contribute. Total lines = Lines without restrictions - lines formed by collinear points + one line Total lines = 105 - 10 + 1 Total lines = 96 Hence, the correct answer is (3) 96.

Key concepts

Combination in Permutation and Combination

One of the fundamental concepts in mathematics, particularly in the study of permutations and combinations, is the idea of a combination. A combination is a selection of items from a larger set, where the order of selection does not matter. This concept is often used in situations where you need to determine the number of possible groups or subsets that can be formed from a given set of distinct elements.

For example, if you have a set of letters {A, B, C}, the possible combinations of two letters are AB, AC, and BC. Notice that BA is not listed as a separate combination since AB and BA contain the same items, just in a different order. In mathematics, we often use the combination formula to calculate this, which is expressed as:
\[ C(n, r) = \frac{n!}{r!\times(n-r)!} \]
Where:

• \( n \) represents the total number of items in the set,
• \( r \) represents the number of items to select,
• \( n! \) is the factorial of n, which is the product of all positive integers up to n,
• \( r! \) is the factorial of r, and
• \( (n-r)! \) is the factorial of the difference between n and r.

The factorial function grows very fast, which means that the number of combinations can become quite large even for small values of n and r. This is why combination problems often require good problem-solving strategies and a methodical approach to avoid errors and oversights in calculations.

Mathematical Problem-Solving

Mathematical problem-solving is an essential skill that involves understanding the problem, devising a plan, carrying out the plan, and then looking back to check and interpret the results. In the context of solving combination problems, this process can be broken down into systematic steps, as in the provided exercise.

Initially, the problem requires comprehension; identify what is given and what is asked for. In our exercise, we were given a set of 15 points and the collinearity condition for 5 points. The next phase, planning, involves realizing that we can use the combination formula to calculate the number of straight lines. The execution step has us perform the calculation for two-point combinations and adjust for the collinear restriction. Finally, reviewing involves checking our work for accuracy, ensuring we didn't miss any points or possible lines, and validating that our approach correctly solves the original problem. It's this reflective practice that often solidifies understanding and improves mathematical thinking for future problems.

Concept of Collinearity in Geometry

Collinearity in geometry refers to the property of points lying on the same straight line. When points are collinear, they form a line segment or a continuous line if there are more than two points. This concept is integral to many geometric problems and can sometimes add further complexity, as it did in our exercise.

Understanding the significance of collinearity allows us to approach problems with greater insight. In our exercise, we had to consider not only the standard combination calculation for the total number of lines from 15 points but also account for the fact that 5 of these points are collinear. As a result, these 5 points only contribute one line rather than the several lines that non-collinear points would contribute. Recognizing such conditions in geometric problems is key to accurate and efficient problem-solving. Accounting for collinearity prevented us from overcounting the lines and helped arrive at the correct solution.