A rectangle ABCD must be constructed on the xy plane so that the side

Question

A rectangle ABCD must be constructed on the xy plane so that the side AB is parallel to the y axis. If the x and y coordinates of A, B, C, and D are integers between (-3) and 6, inclusively, how many different rectangles can be constructed that satisfy these properties?

A. 81
B. 100
C. 2,025
D. 10,000
E. 12,100

A. 81
B. 100
C. 2,025
D. 10,000
E. 12,100

KarishmaB · Accepted Answer

x and y co-ordinates are integers between -3 and 6 so you have 10 values for each. Note how you make a rectangle such that its side is parallel to y axis. Its other side will be parallel to x axis.

The co-ordinates are interdependent as shown. So what you need is two distinct values for m and q (two values from the 10 values ranging from -3 to 6) and two distinct values for n and p (two values from the 10 values ranging from -3 to 6).

So 10C2 makes you pick two values for x co-ordinates m and q
Another 10C2 makes you pick two values for y co-ordinates n and p.
Overall you get 10C2 * 10C2 = 2025 ways

Ichiro49 · Answer

My answer is 2025. 
Because 10C2 * 10C2=2025
But I can't find the number above among options, though C, 2050 is showed as the correct answer.
Please let me know the process through which this problem is soleved.

righthand · Answer

I agree with  Ichiro49 , there are 10 horizontal lines, and 10 vertical lines as it is given that one side is parallel with Y- axis.

So, ans should be  10C2*10C2 = 2025.

DisciplinedPrep · Answer

Ichiro49   righthand  - Thanks for highlighting the issues; I have fixed the answer choices.

ishitaz · Answer

Hi, can someone help me? Shouldn't it be 10C4? 
Let's say A is (x,y) then B will be (x, y') 
=> C will be (x',y) and D will be (x',y')

x,y,x',y' form coordinates in one case i.e 4
Total possible values = 10

Therefore, total no. of ways 10C4?

Thanks in advance,
Ishita

righthand · Answer

Interesting. Let's see, if we continue from x,y,x',y'. We have 10 values to chose from(-3 to 6) for X-coordinate and Y-coordinates. Now, as you have already noted down for X-coordinate we need to chose 2-values (one for x, other for x') among four co-ordinates. No. of ways to do it = 10C2 . Similarly, for Y-coordinate also = 10C2 Hence total rectangles that can be formed = 10C2*10C2. Now, the method that I followed: You need 2 horizontal lines and 2 Vertical lines to create rectangle. Here with co-ordinate values given from -3 to 6, you can create 10 horizontal lines & 10 vertical lines. You need to choose 2 horizontal lines from 10 and 2 vertical lines from 10. Hence answer = 10C2 * 10C2. Hope this helps

TheBunk · Answer

Hi,

I don't understand something. Once you have chosen 3 out of the 4 coordinates, you don't have a choice for the last one, otherwise it won't be a rectangle. So why do we say the answer is 10C2*10C2 ?

I think that you have 10 choice for the first coordinate, then 9 choice for the coordinate on the same line; now you need one more coordinate in on the same line as one of the other 2, so 9 possibilities and then you don't have a choice for the last one.
So you have: 10*9*9 possibilities.

Please tell me where I am wrong.

VeritasKarishma   Bunuel

TheBunk · Answer

Great explanation! Thanks!

harshsahay · Answer

Hi,
Thank you for the explanation. However, shouldn't we multiply 2025 with 4 (as there would be 4 variations of ABCD Rectangle taking AB || y-axis). Thus the answer should be 12100 (E). 
Kindly clarify  VeritasKarishma

KarishmaB · Answer

All variations are included in 10C2 * 10C2. Give me an example of a rectangle and the 3 variations you think are not included. I will give you the values of the co-ordinates which include them.

harshsahay · Answer

Hi, 
Let's take a rectangle with co-ordinates {(1,1),(1,-1),(-1,-1),(-1,1)}.Taking AB || y-axis, we can create four rectangles as follows:
1. A:(1,1), B:(1,-1), C:(-1,-1), D:(-1,1)
2. A:(1,-1), B:(1,1), C:(-1,1), D:(-1,-1)
3. A:(-1,1), B:(-1,-1), C:(1,-1), D:(1,1)
4. A:(-1,-1), B:(-1,1), C:(1,1), D:(1,-1)

There will be four variations for all such 10C2 * 10C2 rectangles. I mean I get it that they represent the same co-ordinates and are placed at the same position with respect to each other and the x-y axis, but with different names, shouldn't the rectangles be considered different.
Or am I thinking too much?
A clarification would be very helpful  VeritasKarishma  ma'am.
Thank you.

KarishmaB · Answer

I get your point. Note that two rectangles that overlap (all four co-ordinates are the same) are the same rectangle, no matter how you name them. We cannot consider them distinct.

shlok11 · Answer

VeritasKarishma  mam,
why can't we solve this question as- we can create a rectangle by having the diagonally opposite coordinates 
so if we have 10 possible values of each X and Y then the number of rectangles will be - possible values of first X coordinate*possible values of second X coordinate *possible values of first Y coordinate * possible values of second Y coordinates=10*9*10*9=8100

shlok11 · Answer

Hello  VeritasKarishma  mam,
sorry for the double post . My question is also that since we are given the names of the coordinates shouldn't we consider the rectangles with the same coordinate values but different coordinate names as different rectangles?

EeshitaV · Answer

Can someone please address this?

Thanks!

himanshu0077 · Answer

It will be 8100/4 = 2025. I also got 8100 as the answer but by different method. But as its already pointed out, ordering of vertices do not matter (cannot comment on this, according to me it should), so answer is 2500.

devendra.c · Answer

This won't allow for the selection of the rectangle which has a point (1,1) and other similar coordinates where the value of x and y is equal

.

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A rectangle ABCD must be constructed on the xy plane so that the side

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