SOLUTION: Four lines on a plane have at most N common points. FIND N. A)7 B)6 C)5 D)4

Question 1143914: Four lines on a plane have at most N common points. FIND N.
A)7
B)6
C)5
D)4
Found 3 solutions by MathLover1, Alan3354, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

First, you have straight lines. They would have at most common point.
Next, add one more line. This new line can intersect both existing lines, adding points to the set of common points.
Finally, add one more line. This new line can intersect all existing lines, adding points to the set of common points.
So in total, we have .
answer:
B)
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
Four lines on a plane have at most N common points. FIND N.
===========
Not a well posed problem.
---
If 2 or more or the lines are coincident, N = an infinite number.
Not more than that.
Answer by ikleyn(52798)   (Show Source): You can put this solution on YOUR website!
.

            More accurate formulation is THIS :

Four lines on a plane have at most N common points. FIND N.

Solution

Two distinct straight lines may have at most 1 common point (the intersection point). So the maximum number of common points of " n " lines is equal to the number of all pairs of lines of the given set of lines. More exactly, the maximum number of common points of N lines is equal to the number of all unordered pairs of lines of the given set of lines. In other words, the maximum number of common points of N lines is equal to the number of all combinations of given lines taken 2 at a time = . In case n = 4, the maximum number of common points of lines is = = 2*3 = 6. This maximum number is achieved if and only if any two lines from the set intersect each other, i.e. are not parallel.

Solved, explained, answered and completed.

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I will give a BONUS problem to you to develop your mind.

Find the maximum possible number of intersection points that N circles may have on a plane ?

If you understand my solution to the previous problem, you should be able to solve this one, too.

-------------------

On Combinations, see introductory lessons
    - Introduction to Combinations
    - PROOF of the formula on the number of Combinations
    - Problems on Combinations
    - OVERVIEW of lessons on Permutations and Combinations
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Combinatorics: Combinations and permutations".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.

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