Question 200762: Okay I need help understanding how many times 10 lines would intersect each other. For instance 2 lines intersect once how about the rest? How does this all work?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Think of the problem like this
Let
L1 = Line #1
L2 = Line #2
L3 = Line #3
L4 = Line #4
L5 = Line #5
L6 = Line #6
L7 = Line #7
L8 = Line #8
L9 = Line #9
L10 = Line #10
Now you want to know the maximum number of intersections between lines 1 through 10. So you'll have the intersections between L1 and L2, between L2 and L3, L4 and L5, etc. We'll denote these intersections as the ordered pairs (L1, L2), (L2, L3), (L3, L4), (L4, L5), etc...
So you'll have the ordered pairs (these are ALL the possible intersections of the 10 lines):
| (L1, L2), | (L1, L3), | (L1, L4), | (L1, L5), | (L1, L6), | (L1, L7), | (L1, L8), | (L1, L9), | (L1, L10) | | (L2, L1), | (L2, L3), | (L2, L4), | (L2, L5), | (L2, L6), | (L2, L7), | (L2, L8), | (L2, L9), | (L2, L10) | | (L3, L1), | (L3, L2), | (L3, L4), | (L3, L5), | (L3, L6), | (L3, L7), | (L3, L8), | (L3, L9), | (L3, L10) | | (L4, L1), | (L4, L2), | (L4, L3), | (L4, L5), | (L4, L6), | (L4, L7), | (L4, L8), | (L4, L9), | (L4, L10) | | (L5, L1), | (L5, L2), | (L5, L3), | (L5, L4), | (L5, L6), | (L5, L7), | (L5, L8), | (L5, L9), | (L5, L10) | | (L6, L1), | (L6, L2), | (L6, L3), | (L6, L4), | (L6, L5), | (L6, L7), | (L6, L8), | (L6, L9), | (L6, L10) | | (L7, L1), | (L7, L2), | (L7, L3), | (L7, L4), | (L7, L5), | (L7, L6), | (L7, L8), | (L7, L9), | (L7, L10) | | (L8, L1), | (L8, L2), | (L8, L3), | (L8, L4), | (L8, L5), | (L8, L6), | (L8, L7), | (L8, L9), | (L8, L10) | | (L9, L1), | (L9, L2), | (L9, L3), | (L9, L4), | (L9, L5), | (L9, L6), | (L9, L7), | (L9, L8), | (L9, L10) | | (L10, L1), | (L10, L2), | (L10, L3), | (L10, L4), | (L10, L5), | (L10, L6), | (L10, L7), | (L10, L8), | (L10, L9), |
Note: one ordered pair (L8, L4) means that you have the intersection between Lines 8 and 4.
Now count all the ordered pairs to get 90 ordered pairs total.
Now if you look through the list, you'll notice that the ordered pairs (L1, L2) and (L2, L1) are the same intersection. So this means that there's twice as much intersections than there really are. So divide the count of 90 by 2 to get 45.
So there are 45 different intersections (maximum) with 10 distinct lines.
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If this didn't make sense, or if you want the shortcut, then simply compute the combination 10 C 2 to get
10 C 2 = (10!)/(2!(10-2)!) = (10!)/(2!(8)!) = (10*9)/2 = 90/2 = 45
So you still get 45 different intersections.
If you are really bold (and really careful), you can draw out the 10 lines and count the number of intersections. If not, then try drawing out 5 lines and test the formula.
If you have any questions, email me at jim_thompson5910@hotmail.com.
Check out my website if you are interested in tutoring.
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