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

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

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New post 27 Jun 2010, 09:20
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44% (02:00) correct 56% (00:54) wrong based on 27 sessions

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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
[Reveal] Spoiler: OA

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Location: Toronto
Re: Two integers from 1 to 16 are chosen at random and their corresponding [#permalink]

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New post 27 Jun 2010, 10:14
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Nusa84 wrote:
Hello everybody,

Could someone help me solving this? I have searched for it in the forum but I haven't found it.

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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

thanks


Hi!

Let's start with the formula that will govern the question (it's always a good idea to job down relevant formulae on your noteboard so you can see what you need to solve the problem):

Probability = # of desired outcomes/total # of possibilities.

The denominator is the easier part of the fraction to calculate. We have 16 total squares and we're choosing 2 of them, so we use the combinations formula:

16C2 = 16!/2!14! = 16*15/2 = 8*15 = 120

While there may be some fancy way to calculate the # of desired outcomes, i.e. how many pairs of squares actually form a rectangle, good ol' brute force is certainly going to be the quickest.

Looking at the diagram, we see that in each row, there are 3 pairs of rectangle-forming squares (e.g. in row 1, we can choose 1+2, 2+3 or 3+4). Since there are 4 rows, there are 4*3 = 12 horizontal rectangle-forming pairs.

Similarly, there are 3 pairs of rectangle-forming squares in each column (e.g. in column 1, we can choose 1+5, 5+9 or 9+13). Since there are 4 columns, there are 4*3 = 12 vertical rectangle-forming pairs.

So, the total number of rectangle-forming pairs is 12+12 = 24.

Our final calculation:

24/120 = 1/5... choose (D).

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Re: Two integers from 1 to 16 are chosen at random and their corresponding [#permalink]

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New post 01 Dec 2017, 05:53
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Re: Two integers from 1 to 16 are chosen at random and their corresponding   [#permalink] 01 Dec 2017, 05:53
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