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If 4 points are indicated on a line and 5 points are indicat

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If 4 points are indicated on a line and 5 points are indicat [#permalink] New post 08 May 2018, 04:00
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Question Stats:

80% (07:36) correct 20% (01:28) wrong based on 10 sessions
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 indicat [#permalink] New post 08 May 2018, 07:07
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amorphous 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


There are two ways to create these triangles:
1) select 2 points from the first line (the line with 4 points) and select 1 point from the second line (the line with 5 points)
2) select 1 point from the first line (the line with 4 points) and select 2 points from the second line (the line with 5 points)


Once we have selected 3 points, we simply connect them with lines to form a triangle

We'll handle each option separately.

----------------------------------------
1) select 2 points from the first line (the line with 4 points) and select 1 point from the second line (the line with 5 points)
Take the task of selecting points and break it into stages.

Stage 1: Select two points from the first line
Since the order in which we select the points does not matter, we can use combinations.
We can select 2 points from 4 points in 4C2 ways (6 ways)
So, we can complete stage 1 in 6 ways

Stage 2: Select one point from the second line
There are 5 points to choose from, so we can complete this stage in 5 ways.

By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus create triangles) in (6)(5) ways (= 30 ways)
----------------------------------------

2) select 1 point from the first line (the line with 4 points) and select 2 points from the second line (the line with 5 points)
Stage 1: Select one point from the first line
There are 4 points to choose from, so we can complete this stage in 4 ways.

Stage 2: Select two point from the second line
Since the order in which we select the points does not matter, we can use combinations.
We can select 2 points from 5 points in 5C2 ways (10 ways)
So, we can complete stage 2 in 10 ways

By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus create triangles) in (4)(10) ways (= 40 ways)
--------------------------------------------

TOTAL number of triangles = 30 + 40 = 70

Answer: D

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Re: If 4 points are indicated on a line and 5 points are indicat   [#permalink] 08 May 2018, 07:07
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