For the following two (02) items : There are 8 points on a plane out of which 4 points are collinear.
52
To solve this problem, we need to determine how many triangles can be formed by joining a set of given points. According to the information provided, there are 8 points on a plane, out of which 4 are collinear.
Let's break down the solution step by step:
- To form a triangle, we need to select 3 non-collinear points.
- First, find the total number of ways to select 3 points out of 8 points. This can be calculated using the combination formula: \(^{8}C_{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56\).
- Next, we need to subtract the number of triangles that cannot be formed because they fall on the same line. Since there are 4 collinear points, selecting any 3 of these collinear points will not form a triangle. The number of combinations of selecting 3 points from these 4 collinear points is: \(^{4}C_{3} = \frac{4 \times 3 \times 2}{3 \times 2 \times 1} = 4\).
- Therefore, the number of triangles that can be formed is the total combinations of 3 points from 8 points minus the collinear combination: \(56 - 4 = 52\).
Thus, the correct answer is 52. This is the total number of triangles that can be constructed from the given points, considering the collinear constraint.
Correct Option: 52
Important Questions from Permutation and Combination
If Quantity A is the number of ways to assign a number from 1 to 5 without repetition to each of four people, and Quantity B is the number of ways to assign a number from 1 to 5 without repetition to each of 5 people, then which of the following statements is correct with respect to Quantities A and B?
Which of the following muscles regulates the exit of food from the stomach into the small intestine?
There are 9 cups placed on a table arranged in equal number of rows and columns out of which 6 cups contain coffee and 3 cups contain tea. In how many ways can they be arranged so that each row should contain at least one cup of coffee?
In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?
There are $15$ distinct points on a plain sheet of paper. If $4$ of these points are collinear, find the maximum number of triangles that can be drawn using these points.
