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# Two integers from 1 to 16 are chosen at random and their corresponding

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Re: Two integers from 1 to 16 are chosen at random and their corresponding [#permalink]
Nusa84 wrote:
Attachment:
Dibujo.JPG

Two integers from 1 to 16 are chosen at random and their corresponding square in the above grid is shaded. What is the probability that these two shaded squares form a rectangle?

a. 1/10
b. 1/8
c. 1/6
d. 1/5
e. 1/4

Total number of outcomes for selecting 2 numbers = 16 * 15 = 240
As we can select 16 numbers in the first go and 15 remaining numbers in the second.

Preferred Outcomes -
We will need to see the placement of the numbers to see preferred outcomes:
4 corner numbers can be linked with 2 numbers each to form a rectangle. For e.g. 1 can be linked with 2 or 5. Hence 4*2 = 8
8 other numbers along the boundary ( 2, 3, 5, 8, 9, 12, 14, 15) can be linked with 3 numbers each. For e.g. 2 can be linked with 1, 3 or 6. Hence 8*3 = 24
4 numbers in the middle can be linked with 4 numbers each. For e.g. 6 can be linked with 2, 5, 7 or 10. Hence 4*4 = 16

Preferred Outcomes = 8 + 24 + 16 = 48

Probability = 48/240 = 1/5

Hence D is the correct answer.
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Re: Two integers from 1 to 16 are chosen at random and their corresponding [#permalink]
Another way to look at the problem would be:

1) Number of ways to select 2 squares such that they are NOT in the same row or column = (16C1*9C1)/2! = 72 ways
2) Number of ways to select 2 squares such that they are IN the same row or column = 16C2 - 72 = 48 ways
3) We notice that only 1/2 of the time that when the squares are in the same row or column, they are next to each other i.e. they form a rectangle. Therefore, 48/2 = 24 ways.
4) Total probability = 24/120 = 1/5 (Answer: D)
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Re: Two integers from 1 to 16 are chosen at random and their corresponding [#permalink]
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Re: Two integers from 1 to 16 are chosen at random and their corresponding [#permalink]
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