Question 476372: There are 10 points in a plane. No three of these points are in a straight line, except 4 points which are all in the same straight line. How many straight lines can be formed by joining the 10 points?
total no. of points = n= 10
r = 2 (because each line requires 2 points to be created)
10C2 = 45 total lines can be made
Question 1: The total points refer to 4 points that are collinear and 6 points that are just anywhere in space? Is that correct?
4 collinear points will form only 1 line instead of the hypothetical 4C2 = 6 lines
Question 2: Will 4 points form 6 lines only when they are located anywhere in space?
Question 3: Why do I need to subtract 4C2 = 6 lines from 10C2? Is it because the 6 lines are already included in 10C2?
total no. of lines is = 45 - 6 + 1 = 40
I subtracted 6 because the total number of lines created by 10 points includes 6 lines, so 6 is like having doubles.
Question 4: Why 6 lines have to be subtracted from 45 lines?
Question 5: Does 1 line added to 45 lines account for the 1 line formed by 4 collinear points?
Question 6: What does this mean in mathematical language? "No three of these points are in a straight line, except 4 points which are all in the same straight line."
Here is the textbook's answer:
Number of lines formed if no 3 of the 10 points were in a straight line = 10C2 = 45.
Question 7: I can hardly understand what it means; to me it's just means total number of lines.
Number of lines formed by 4 points, no 3 of which are collinear=4C2=6
Question 8: I cannot really understand what "no 3 of which are collinear" really mean when translated from the language of mathematics to the word problem language.
Since the four points are collinear they form 1 line instead of 6 lines.
Required number of lines = 45-6+1 = 40 lines.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! You are making this question way too complicated. If your textbook is asking all those questions then I will presume that the textbook is making this question too complicated.
I think I have already solved this problem a few days ago, but I will solve it again for clarification. We form lines by choosing pairs of points (since two points uniquely determine a line). We have three cases:
Case 1: We choose two of the four collinear points. Here, only one line is determined because we cannot count 4C2 lines (they are all the same).
Case 2: We choose two of the six other non-collinear points. Here, it is simply choosing any two points out of 6, 6C2 = 15.
Case 3: We choose one of the four collinear points, and one of the six remaining points. There are 4*6 = 24 lines determined, no two of which are collinear.
The number of lines is 1+15+24 = 40.
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Okay, now for your "questions." #1 - yes. #2 - yes, we choose two points from four, 4C2 = 6. #3 and #4 are essentially the same. You subtract 4C2 = 6 because you are counting the same line six times, when you only want to count it once (hence, add 1 afterwards). #5 - yes, just did that. #6 - This means that you have four points that lie on the same line, and the other six are completely random and do not lie on this line. #7 - Yes, correct? #8 - "Collinear" means "on the same line."
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