A rectangle ABCD is to be constructed on the XY plane

Question

A rectangle ABCD is to be constructed on the XY plane so that AB is || to the y axis ; if the X and Y co-ordinates of A,B,C & D are integers between -3 to 6 ; How many different rectangles can be constructed ?

A > 81
B > 100
C > 2025
D > 10000
E > 12100

A) 81
B) 100
C) 2025
D) 10000
E) 12100

chetan2u · Accepted Answer

So we have X and Y ranging from -3 to 6 that is 10 by 10 integer values. 
You can choose 2 points in X coordinates in 10C2 ways, and for each of this way, we can choose 2 points in y coordinates in 10C2 ways. 
Thus total ways = 10C2*10C2=45*45=2025

C

To understand the concept of 10C2*10C2, I have taken a 3 by 3 rectangle in which the answer will be 3C2*3C2=3*3=9

gvij2017

peace31 · Answer

Thanks for bringing up this triangle question  zs2 . So the basic difference between the rectangle question and triangle question is the usage of the permutations concept, thereby reflecting in the formula to solve these individual questions.

In the rectangle question, we are choosing the two points on each axis (x1, x2 or y1, y2) at the same time. Thus we are opting for the combination formula nCr. The shape of the rectangle doesn't change when we swap x1 with x2 or y1 with y2. Hence the whole number of cases should be divided by 2 in order to eliminate the repeating cases. Hence (10x9)/2 which is nothing but 10C2.

In case of triangle question, we are choosing the points in similar fashion (x1, x2 or y1, y2) except that if we swap the points the shape of the triangle changes. Hence there is no need to divide the whole cases by 2. Since the order of the chosen points matter here, we use permutation formula which is nPr. Hence (11x10) which is nothing 11P10.

Hope this helped.

gvij2017 · Answer

Could you elaborate 10C2*10C2?
According to my understanding for your explanation, if I choose any two x coordinates from 10 x coordinates and two y coordinates from 10 y coordinates, then I could other shapes such as quadrilaterals, two intersecting lines as well.
Please help to understand this concept.

Peddi · Answer

I didnt understand your approach , can you elaborate

Posted from my mobile device

CareerGeek · Answer

A rectangle can be formed by selecting any 2 values of x and any 2 values of y

E.g: If we select  x_1, x_2, y_1  &  y_2 , a rectangle can be formed with coordinates  (x_1, y_1), (x_1, y_2), (x_2, y_1)  &  (x_2, y_2)

--> Number of different rectangles that can be formed = Number of ways of selecting 2 values of x & 2 values of y out of 10 values =  10c_2*10c_2  =  45*45 = 2025

IMO Option C

gvij2017 · Answer

I appreciate your elucidation. Actually I also missed the point that AB is parallel to Y axis.
In that case I would need 2 x coordinates and 2 y coordinates. 
Arrangement doesn't matter here so I have to use combination concept.

There is one new learning from your every explanation. 
Thanks!

unraveled · Answer

Peddi Tried as concise an explanation as it can be but this is best i can offer. You have to be a little patient.

The coordinates within which these rectangles can be formed are (-3,6) (6,6) (6,-3) (-3,-3). From Point no. 1 to 9 = 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 285 From Point No. 10 to 17 = 2 * (72 + 63 + 54 + 45 + 36 + 27 + 18 + 9) = 648 From Point No. 18 to 24 = 2 * (56 + 48 + 40 + 32 + 24 + 16 + 8) = 448 From Point No. 25 to 30 = 2 * (42 + 35 + 28 + 21 + 14 + 7) = 294 From Point No. 31 to 35 = 2 * (30 + 24 + 18 + 12 + 6) = 180 From Point No. 36 to 39 = 2 * (20 + 15 + 10 + 5) = 100 From Point No. 40 to 42 = 2 * (12 + 8 + 4) = 48 From Point No. 43 to 44 = 2 * (6 + 3) = 18 From Point No. 45 = 2 * (2) = 4 Total formations are = 285 + 648 + 448 + 294 + 180 + 100 + 48 + 18 + 4 = 2025. Explanation as below: (refer figure) Rectangle .JPG So, we can take rectangles(including squares) with dimensions as follows: 1. 1Hx1V (sides with two consecutive coordinates for example (-3,6) (-2,6) (-2,5) (-3,5) and so on). Total formations = 9 x 9 = 81 2. 2Hx2V (sides with two coordinates having 1 coordinate in between for example (-3,6) (-1,6) (-1,4) (-3,4) and so on) Total formations = 8*8 = 64 3. 3Hx3V (sides with two coordinates having 2 coordinates in between for example (-3,6) (0,6) (0,3) (-3,3) and so on) Total formations = 7*7 = 49 4. 4Hx4V (sides with two coordinates having 3 coordinates in between for example (-3,6) (1,6) (1,2) (-3,2) and so on) Total formations = 6*6 = 36 5. 5Hx5V (sides with two coordinates having 4 coordinates in between for example (-3,6) (2,6) (2,1) (-3,1) and so on) Total formations = 5*5 = 25 6. 6Hx6V (sides with two coordinates having 5 coordinates in between for example (-3,6) (3,6) (3,0) (-3,0) and so on) Total formations = 4*4 = 16 7. 7Hx7V (sides with two coordinates having 6 coordinates in between for example (-3,6) (4,6) (4,-1) (-3,-1) and so on) Total formations = 3*3 = 9 8. 8Hx8V (sides with two coordinates having 7 coordinates in between for example (-3,6) (5,6) (5,-2) (-3,-2) and so on) Total formations = 2*2 = 4 9. 9Hx9V (sides with two coordinates having 8 coordinates in between for example (-3,6) (6,6) (6,-3) (-3,-3) and so on) Total formations = 1*1 = 1 (where H and V are just for horizontal and vertical reference of side dimensions) Up till now all formations were of square form. Other rectangles are: (refer figure) 10. 1Vx2H (sides with two coordinates having 0 coordinates and having 1 coordinate along y-axis and x-axis respectively in between for example (-3,6) (-1,6) (-1,5) (-3,5) and so on) Total formations = 9 x 8 = 72 11. 1Vx3H (sides with two coordinates having 0 coordinates and having 2 coordinates along y-axis and x-axis respectively in between for example (-3,6) (0,6) (0,5) (-3,5) and so on) Total formations = 9 x 7 = 63 12. 1Vx4H (sides with two coordinates having 0 coordinates and having 3 coordinates along y-axis and x-axis respectively in between for example (-3,6) (1,6) (1,5) (-3,5) and so on) Total formations = 9 x 6 = 54 13. 1Vx5H (sides with two coordinates having 0 coordinates and having 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,5) (-3,5) and so on) Total formations = 9 x 5 = 45 14. 1Vx6H (sides with two coordinates having 0 coordinates and having 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,5) (-3,5) and so on) Total formations = 9 x 4 = 36 15. 1Vx7H (sides with two coordinates having 0 coordinates and having 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,5) (-3,5) and so on) Total formations = 9 x 3 = 27 16. 1Vx8H (sides with two coordinates having 0 coordinates and having 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,5) (-3,5) and so on) Total formations = 9 x 2 = 18 17. 1Vx9H (sides with two coordinates having 0 coordinates and having 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,5) (-3,5) and so on) Total formations = 9 x 1 = 9 18. 2Vx3H (sides with two coordinates having 1 coordinate and 2 coordinates along y-axis and x-axis respectively in between for example (-3,6) (0,6) (0,4) (-3,4) and so on) Total formations = 8 x 7 = 56 19. 2Vx4H (sides with two coordinates having 1 coordinate and 3 coordinates along y-axis and x-axis respectively in between for example (-3,6) (1,6) (1,4) (-3,4) and so on) Total formations = 8 x 6 = 48 20. 2Vx5H (sides with two coordinates having 1 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,4) (-3,4) and so on) Total formations = 8 x 5 = 40 21. 2Vx6H (sides with two coordinates having 1 coordinate and 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,4) (-3,4) and so on) Total formations = 8 x 4 = 32 22. 2Vx7H (sides with two coordinates having 1 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,4) (-3,4) and so on) Total formations = 8 x 3 = 24 23. 2Vx8H (sides with two coordinates having 1 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,4) (-3,4) and so on) Total formations = 8 x 2 = 16 24. 2Vx9H (sides with two coordinates having 1 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,4) (-3,4) and so on) Total formations = 8 x 1 = 8 Again 25. 3Vx4H (sides with two coordinates having 2 coordinate and 3 coordinates along y-axis and x-axis respectively in between for example (-3,6) (1,6) (1,3) (-3,3) and so on) Total formations = 7 x 6 = 42 26. 3Vx5H (sides with two coordinates having 2 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,3) (-3,3) and so on) Total formations = 7 x 5 = 35 27. 3Vx6H (sides with two coordinates having 2 coordinate and 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,3) (-3,3) and so on) Total formations = 7 x 4 = 28 28. 3Vx7H (sides with two coordinates having 2 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,3) (-3,3) and so on) Total formations = 7 x 3 = 21 29. 3Vx8H (sides with two coordinates having 2 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,3) (-3,3) and so on) Total formations = 7 x 2 = 14 30. 3Vx9H (sides with two coordinates having 2 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,3) (-3,3) and so on) Total formations = 7 x 1 = 7 31. 4Vx5H (sides with two coordinates having 3 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (2,6) (2,2) (-3,2) and so on) Total formations = 6 x 5 = 30 32. 4Vx6H (sides with two coordinates having 3 coordinate and 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,2) (-3,2) and so on) Total formations = 6 x 4 = 24 33. 4Vx7H (sides with two coordinates having 3 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,2) (-3,2) and so on) Total formations = 6 x 3 = 18 34. 4Vx8H (sides with two coordinates having 3 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,2) (-3,2) and so on) Total formations = 6 x 2 = 12 35. 4Vx9H (sides with two coordinates having 3 coordinate and 4 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,2) (-3,2) and so on) Total formations = 6 x 1 = 6 36. 5Vx6H (sides with two coordinates having 4 coordinate and 5 coordinates along y-axis and x-axis respectively in between for example (-3,6) (3,6) (3,1) (-3,1) and so on) Total formations = 5 x 4 = 20 37. 5Vx7H (sides with two coordinates having 4 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,1) (-3,1) and so on) Total formations = 5 x 3 = 15 38. 5Vx8H (sides with two coordinates having 4 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,1) (-3,1) and so on) Total formations = 5 x 2 = 10 39. 5Vx9H (sides with two coordinates having 4 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,1) (-3,1) and so on) Total formations = 5 x 1 = 5 40. 6Vx7H (sides with two coordinates having 5 coordinate and 6 coordinates along y-axis and x-axis respectively in between for example (-3,6) (4,6) (4,0) (-3,0) and so on) Total formations = 4 x 3 = 12 41. 6Vx8H (sides with two coordinates having 5 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,0) (-3,0) and so on) Total formations = 4 x 2 = 8 42. 6Vx9H (sides with two coordinates having 5 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,0) (-3,0) and so on) Total formations = 4 x 1 = 4 43. 7Vx8H (sides with two coordinates having 6 coordinate and 7 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (5,-1) (-3,-1) and so on) Total formations = 3 x 2 = 6 44. 7Vx9H (sides with two coordinates having 6 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (6,6) (6,-1) (-3,-1) and so on) Total formations = 3 x 1 = 3 45. 8Vx9H (sides with two coordinates having 7 coordinate and 8 coordinates along y-axis and x-axis respectively in between for example (-3,6) (5,6) (6,0) (-3,0) and so on) Total formations = 2 x 1 = 2 Since point no. 10 to 45 will have a mirror image along the diagonal (-3,6) – (6,-3), the number of each formations calculated above for point no. 10 to 45 would be doubled. Example: For Point No. 10, 1Vx2H (sides with two coordinates having 0 coordinates and having 1 coordinate along y-axis and x-axis respectively in between for example (-3,6) (-1,6) (-1,5) (-3,5)) would have an exact replica as 2Vx1H (sides with two coordinates having 0 coordinates and having 1 coordinate along y-axis and x-axis respectively in between for example (-3,6) (-2,6) (-2,4) (-3,4)) and so on). Then Total formations would be = 2 x 9 x 8 = 144 and so on… gvij2017 Have a look. Since side of rectangle is always || to an axis, only these are possible formations. Hope this is helpful.

zs2 · Answer

HI

Can anyone tell the difference between this question and Triangle question.

Why cant solution to this question be --- 10C1 * 9C1 * 10C1*9C1 like in below question ---

Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and Y coordinates of P,Q and R are to be integers that satisfy the inequalitites -4≤ X≤ 5 and 6≤ y≤ 16. How many different triangles with these properties could be constructed?

A)110
B)1100
C)9900
D)10000
E)12100

thanks..

We have the rectangle with dimensions 10*11 (10 horizontal dots and 11 vertical). PQ is parallel to y-axis and PR is parallel to x-axis.

Choose the (x,y) coordinates for vertex P (right angle): 10C1*11C1;
Choose the x coordinate for vertex R (as y coordinate is fixed by A): 9C1, (10-1=9 as 1 horizontal dot is already occupied by A);
Choose the y coordinate for vertex Q (as x coordinate is fixed by A): 10C1, (11-1=10 as 1 vertical dot is already occupied by A).

10C1*11C1*9C1*10C1=9900.

neelugarg33 · Answer

So we have X and Y ranging from -3 to 6 that is 10 by 10 integer values. 
You can choose 2 points in X coordinates in 10C2 ways, and for each of this way, we can choose 2 points in y coordinates in 10C2 ways. 
Thus total ways = 10C2*10C2=45*45=2025

C

To understand the concept of 10C2*10C2, I have taken a 3 by 3 rectangle in which the answer will be 3C2*3C2=3*3=9

Hi  chetan2u ,

The question says inteer  BETWEEN  -3 to 6.
Should we not exclude -3 and 6 and use 8C2 ways instead of 10C2?

mimishyu · Answer

haven’t thought of this before  neelugarg33  mention up, I just thought the quickest and easiest way to solve this question is by using combination in that order doesn’t matter here, we have 10C2 (choose two points randomly from points -3 -2 -1 0 1 2 3 4 5 6 these 10 objects) ways to choose the two points on x coordinate as well as y coordinate so the total ways to construct a rectangle equals ((10*9)/2)* ((10*9)/2) =45*45=2,025
now I have the same doubt as him since I google on the internet someone also ask “is between inclusive in math”?
the answer is illustrated as thus: there are two integers between 1 and 4, “between” means the space between only. The question will say “inclusive” if a and b are meant to be considered…..
so shouldn't the answer to this question correct as (8C2) * (8C2)

LakshitPareek · Answer

hi, the question says, that integers b/w -3 to 6. why have we taken   and   also. Why are we not taking the middle values (-2,-1,0,1,2,3,4,5. Can someone please explain.

thanks in advance.

Crytiocanalyst · Answer

The traingleshould have coordinates (a,b ) , ( d,b ) , (d,c) ,(a,c)

Therefore we have to chose a,d for x- coordinate and b,c for y-coordinate

This can be done in 10C2 * 10C2 ways = 2025

Therefore IMO C

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A rectangle ABCD is to be constructed on the XY plane

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A rectangle ABCD is to be constructed on the XY plane

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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