GMAT question - The coordinates (x,y) of each corners of rectangle A... - Review

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Type: Problem Solving

Difficulty: 5 star rating

The coordinates (x,y) of each corners of rectangle ABCD are such that x and are integers and satisfy the equations `2 <= x <= 5` and `-2 <= y <= 2`. The edges of the rectangle are parallel to the X and Y axes. How many distinct rectangles could be formed which would satisfy these requirements?

  1. 240
  2. 60
  3. 20
  4. 18
  5. 9

Explanation

In this question you are being asked to find the number of distinct selections of coordinates that can be made.

The key selections are the x coordinates of the left and right hand edges and the y coordinates of the top and bottom edges.

You need to select 2 x coordinates from a possible 4 for the left and right edges and 2 y coordinates from a possible 5 for the top and bottom edges.

Since order does not matter in this case we should use the combinations formula to work out the number of possibilities.

`C = (n!)/(r!(n-r)!)`

For the x coordinates it is selecting 2 from 4 so

`text{Number of x coordinate selections is} = (4!)/(2!(4-2)!)`

`text{Number of x coordinate selections is} = (4!)/(2!2!)`

`text{Number of x coordinate selections is} = (4 xx 3 xx 2 xx 1)/(2 xx 1 xx 2 xx 1)`

`text{Number of x coordinate selections is} = 6`

For the y coordinates it is selecting 2 from 5 so

`text{Number of y coordinate selections is} = (5!)/(2!(5-2)!)`

`text{Number of y coordinate selections is} = (5!)/(2!3!)`

`text{Number of y coordinate selections is} = (5 xx 4 xx 3 xx 2 xx 1)/(2 xx 1 xx 3 xx 2 xx 1)`

`text{Number of y coordinate selections is} = 10`

Therefore the total number of possible rectangles is 6 × 10 = 60 and the correct answer is C.

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