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If 4 points are indicated on a line and 5 points are indicated on
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16 May 2012, 06:52
Question Stats:
74% (01:37) correct 26% (01:51) wrong based on 452 sessions
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If 4 points are indicated on a line and 5 points are indicated on another line that is parallel to the first line, how many triangles can be formed whose vertices are among the 9 points? A. 20 B. 30 C. 40 D. 70 E. 90
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Re: If 4 points are indicated on a line and 5 points are indicated on
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16 May 2012, 07:58
alexpavlos wrote: If 4 points are indicated on a line and 5 points are indicated on another line that is parallel to the first line, how many triangles can be formed whose vertices are among the 9 points?
A. 20 B. 30 C. 40 D. 70 E. 90
Tried doing a few options, but couldn't get it right and the answer provided left me more confused! Approach #1:There are two types of triangles possible: With two vertices on the line with 4 points and the third vertex on the line with 5 points > \(C^2_4*C^1_5=30\); With two vertices on the line with 5 points and the third vertex on the line with 4 points > \(C^2_5*C^1_4=40\); Total: \(30+40=70\). Answer: D. Approach #2:All different 3 points out of total 4+5=9 points will create a triangle EXCEPT those 3 points which are collinear. \(C^3_{9}(C^3_4+C^3_5)=84(4+10)=70\) (where \(C^3_4\) and \(C^3_4\) are # of different 3 collinear points possible from the line with 4 points and the line with 5 points, respectively). Answer: D. Hope it's clear.
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Re: If 4 points are indicated on a line and 5 points are indicated on
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16 May 2012, 07:04
say line A with 4 points and line B with 5 points. and triangle is PQR. If P is on line A and QR on line B we can have 4C1*5C2=40 triangles. If P is on line B and QR on line A we can have 4C2*5C1=30 triangles. Total =70 triangles.




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Re: If 4 points are indicated on a line and 5 points are indicated on
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29 Dec 2012, 01:27
Not that fun this question I wonder how often I would see such a question on test day. Solution: * * * * * (line B) * * * * (line A) Choose 2 points from line A and a point from line B: = 4!/2!2! * 5!/1!4! = 30 Choose 1 point from line A and 2 points from line B: = 4!/1!3! * 5!/2!3! = 4 * 10 = 40 Combine possiblities: 30 + 40 = 70
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Re: If 4 points are indicated on a line and 5 points are indicated on
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19 Jul 2014, 21:45
See Image below for what my scratch paper looked like on this one
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Re: If 4 points are indicated on a line and 5 points are indicated on
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19 Jul 2014, 22:47
Bunuel wrote: alexpavlos wrote: If 4 points are indicated on a line and 5 points are indicated on another line that is parallel to the first line, how many triangles can be formed whose vertices are among the 9 points?
A. 20 B. 30 C. 40 D. 70 E. 90
Tried doing a few options, but couldn't get it right and the answer provided left me more confused! Approach #1:There are two types of triangles possible: With two vertices on the line with 4 points and the third vertex on the line with 5 points > \(C^2_4*C^1_5=30\); With two vertices on the line with 5 points and the third vertex on the line with 4 points > \(C^2_5*C^1_4=40\); Total: \(30+40=70\). Answer: D. Approach #2:All different 3 points out of total 4+5=9 points will create a triangle EXCEPT those 3 points which are collinear. \(C^3_{9}(C^3_4+C^3_5)=84(4+10)=70\) (where \(C^3_4\) and \(C^3_4\) are # of different 3 collinear points possible from the line with 4 points and the line with 5 points, respectively). Answer: D. Hope it's clear. I really like your 2nd approach
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Re: If 4 points are indicated on a line and 5 points are indicated on
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06 Aug 2015, 14:54
Bunuel wrote: alexpavlos wrote: If 4 points are indicated on a line and 5 points are indicated on another line that is parallel to the first line, how many triangles can be formed whose vertices are among the 9 points?
A. 20 B. 30 C. 40 D. 70 E. 90
Tried doing a few options, but couldn't get it right and the answer provided left me more confused! Approach #1:There are two types of triangles possible: With two vertices on the line with 4 points and the third vertex on the line with 5 points > \(C^2_4*C^1_5=30\); With two vertices on the line with 5 points and the third vertex on the line with 4 points > \(C^2_5*C^1_4=40\); Total: \(30+40=70\). Answer: D. Approach #2:All different 3 points out of total 4+5=9 points will create a triangle EXCEPT those 3 points which are collinear. \(C^3_{9}(C^3_4+C^3_5)=84(4+10)=70\) (where \(C^3_4\) and \(C^3_4\) are # of different 3 collinear points possible from the line with 4 points and the line with 5 points, respectively). Answer: D. Hope it's clear. Thanks Bunuel. That was great. Do you by any chance have other problems similar to this that you know of?



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Re: If 4 points are indicated on a line and 5 points are indicated on
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06 Aug 2015, 15:29
dukenukem wrote: Bunuel wrote: alexpavlos wrote: If 4 points are indicated on a line and 5 points are indicated on another line that is parallel to the first line, how many triangles can be formed whose vertices are among the 9 points?
A. 20 B. 30 C. 40 D. 70 E. 90
Tried doing a few options, but couldn't get it right and the answer provided left me more confused! Approach #1:There are two types of triangles possible: With two vertices on the line with 4 points and the third vertex on the line with 5 points > \(C^2_4*C^1_5=30\); With two vertices on the line with 5 points and the third vertex on the line with 4 points > \(C^2_5*C^1_4=40\); Total: \(30+40=70\). Answer: D. Approach #2:All different 3 points out of total 4+5=9 points will create a triangle EXCEPT those 3 points which are collinear. \(C^3_{9}(C^3_4+C^3_5)=84(4+10)=70\) (where \(C^3_4\) and \(C^3_4\) are # of different 3 collinear points possible from the line with 4 points and the line with 5 points, respectively). Answer: D. Hope it's clear. Thanks Bunuel. That was great. Do you by any chance have other problems similar to this that you know of? Combination problems are located at search.php?search_id=tag&tag_id=52



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Re: If 4 points are indicated on a line and 5 points are indicated on
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06 Feb 2017, 09:47
Attached is a visual that should help. Bunuel's method is the most elegant, but here is an alternative strategy if the other solution is not clear.
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Re: If 4 points are indicated on a line and 5 points are indicated on
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14 Mar 2018, 16:09
alex1233 wrote: If 4 points are indicated on a line and 5 points are indicated on another line that is parallel to the first line, how many triangles can be formed whose vertices are among the 9 points?
A. 20 B. 30 C. 40 D. 70 E. 90 We have two scenarios: The first scenario is that the 1 vertex is on the first line and the 2 other vertices are on the second line. So the number of ways to create such a triangle is: 4C1 x 5C2 = 4 x (5 x 4)/2 = 40 The second scenario is that 2 vertices are on the first line and the 1 vertex is on the second line. So the number of ways to create such a triangle is: 4C2 x 5C1 = (4 x 3)/2 x 5 = 6 x 5 = 30 Thus, the total number of ways to create the triangle is 40 + 30 = 70. Answer: D
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Re: If 4 points are indicated on a line and 5 points are indicated on
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