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Chapter 5: Problem 401

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

Expert verified
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

01

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.
02

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): \[3 + 21 = 24\]
03

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).

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