Eight points are equally spaced on a circle. If 4 of the 8 points are

Question

Eight points are equally spaced on a circle. If 4 of the 8 points are to be chosen at random, what is the probability that a quadrilateral having the 4 points chosen as vertices will be a square?

A  \frac{1}{70}

B  \frac{1}{35}

C  \frac{1}{7}

D  \frac{1}{4}

E  \frac{1}{2}

BrentGMATPrepNow · Accepted Answer

Let's sketch the 8 points....
 https://i.imgur.com/Zlkaa64.png

Notice that, if we select point A as our first point....
 https://i.imgur.com/Rowiw2c.png

...then, in order to create a square, the other 3 points MUST be C, E and G
 https://i.imgur.com/GXOx4IY.png

Likewise, if we select point B as our first point....
 https://i.imgur.com/Owz4qYQ.png

...then, in order to create a square, the other 3 points MUST be D, F and H
 https://i.imgur.com/zdJJB0f.png

and so on.

So, our first point can be ANY point, but after we've selected the first point, there are exactly 3 points that must be selected in order to create a square.

So....
P(4 points create a square) = P(1st point is ANY point  AND  2nd point is one of the 3 needed points  AND  3rd point is one of remaining points needed  AND  4th point is one of remaining points needed)
= P(1st point is ANY point)  x  P(2nd point is one of the 3 needed points)  x  P(3rd point is one of remaining points needed)  x  P(4th point is one of remaining points needed) 
= 8/8  x  3/7  x  2/6  x  1/5
= 1/35

Answer: B

Cheers,
Brent

Princ · Answer

OA : B

Favourable case : There are only 2 distinct square possible.

Attachment:

Capture.PNG [ 18.42 KiB | Viewed 21895 times ]

One with dotted diagonals and another with solid line diagonals.

Total number of case : \(C(8,4) = \frac{8!}{4!4!}=70\)

Probability that a quadrilateral having the 4 points chosen as vertices will be a square : \(\frac{2}{70}\)=\(\frac{1}{35}\)

EgmatQuantExpert · Answer

Given

• Eight points are equally spaced on a circle.

To Find

• If 4 of the 8 points are to be chosen at random, the probability that a quadrilateral having the 4 points chosen as vertices will be a square.

Approach and Working Out

• Let us denote the points as 1, 2, 3, … 8.
o Total number of ways to select 8 points =  ^8 C 4  = 70

• To make a square we need to select the alternate points.
o Either (1, 3, 5, 7) or (2, 4, 6, 8)

• Answer =  \frac{2}{70}  =  \frac{1}{35}

Correct Answer: Option B

Mofeky · Answer

There are only 2 squares that can be drawn then we divide by the number of possible quadrilaterals (8C4)

Eight points are equally spaced on a circle. If 4 of the 8 points are

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Eight points are equally spaced on a circle. If 4 of the 8 points are

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