A mathematics teacher provides the x- and y- coordinates of 10 points

Question

A mathematics teacher provides the x- and y- coordinates of 10 points in a rectangular coordinate system. These points are (0.5,4),(1,4),(2,4),(3,4),(3,5),(4,5),(5,5),(6,5),(6.5,5) and (7, 5). He asks a student to select 3 points at random. What is the probability that the chosen points form a triangle?

A. 1/9
B. 1/5
C. 2/3
D. 4/5
E. 8/9

A. 1/9
B. 1/5
C. 2/3
D. 4/5
E. 8/9

chetan2u · Accepted Answer

When you look at the coordinates, you should see that y has only 2 values 4 and 5..
So, (0.5,4),(1,4),(2,4),(3,4) are on one line, and other 6 (3,5),(4,5),(5,5),(6,5),(6.5,5) and (7, 5) on other line..
Also both lines are parallel.

We can form a triangle if 2 points are from one line and the third point is on the second line..
so two ways..
1) 2 points on line where y=4....4C2*6C1=36
2) 2 points on line where y=6....6C2*4C1=60
Total 36+60=96

Total ways of choosing 3 points from 10 points = 10C3 =  \frac{10*9*8}{3*2}=120

Thus, probability is  \frac{96}{120}=\frac{4}{5}

D

Archit3110 · Answer

total point 10 
∆ we need 3 points ; 10c3 = 120
y coordiante 4 & 5 are common 
at 4 ; 4 points and 5 ; 6 points
so ∆ can be made ; 4c2*6c1+6c2*4c1 = 96
96/120 ; 4/5
IMO D

singhamit06 · Answer

Total ways of not forming the triangle = 4C3 + 6C3 = 24
Total ways of selecting points = 10C3 = 120
Probability of not forming a triangle when 3 points are selected = 24/120 = 1/5
Probability of forming a triangle = 1-1/5 = 4/5

xyzKunal · Answer

This is a simple counting approach.

First, you should be able to see that first 4 coordinates have y=4 i.e. they are on a single line (y=4)
Similarly, next 6 coordinates have y=5, i.e. they are on a single line (y=5).

To form a triangle, there are two possibilities,
 
 Choose 2 points from line y=4 and 1 point from line y=5 
 4c2 * 6c1 (selecting 2 coordinates out of 4 and 1 out of 6) = 36 
 Choose 1 point from line y=4 and 2 points from line y=5
 4c1 * 6c2 (selecting 1 coordinate out of 4 and 2 out of 6) = 60

total favaourable outcomes = 36+60=90

Total Outcomes = 10C3 = 120

Probability = 96/120 = 4/5
 Option D

_ankitchawla_ · Answer

I went the opposite direction of how others are going.
Notice that first 4 points lie on the same line and last 6 lie on another line.

P(forming a triangle) = 1 - P(not forming a triangle)

So for P(not forming a triangle) :

Total Number of cases = 10C3 = 120
Cases where triangle cannot be formed = Cases where all 3 points chosen, lie on the same line = 6C3 + 4C3 = 24

so final answer =

1/120 - 24/120 = 96/120 = 4/5

Option D

A mathematics teacher provides the x- and y- coordinates of 10 points

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A mathematics teacher provides the x- and y- coordinates of 10 points

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