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If two of the four expressions x+y, x+5y, x-y, and 5x-y are chosen at random, what is the probability that their product will be of the form of x^2 -(by)^2, where b is an integer?

The number of ways we can select 2 out of 4 expressions are 4c2= 6 ways

These are 6 total outcomes possible.
(x+y) (x+5y)
(x+y) (x-y )
(x+y ) (5x-y)
(x+5y) (x-y)
(x+5y) (5x-y)
(x-y) (5x-y)

Out of this outcomes, we should look for the product in form of x^2 - (by)^2 , which is same as (x+by)(x - by), where b is an integer.

We can see that only 1 out of 6 is in the form of (x+by)(x - by) i.e (x+y) (x-y )

So the probability = 1/6
Option E is the answer.

Thanks,
Clifin J Francis,
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GMATNinja, can you please clarify why the following approach is not correct?
According to probability definition, probability=total # of desirable outcomes/total # of possible outcomes.
In this problem, total # of desirable outcomes=2 expressions. 2 expressions from the list can be used to produce the following outcome- x^2 - (by)^2 ; total # of possible outcomes=4 expressions--> Probability that their product will be of the form of x^2 - (by)^2, where b is an integer = 2/4=1/2
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GMATNinja, can you please clarify why the following approach is not correct?
According to probability definition, probability=total # of desirable outcomes/total # of possible outcomes.
In this problem, total # of desirable outcomes=2 expressions. 2 expressions from the list can be used to produce the following outcome- x^2 - (by)^2 ; total # of possible outcomes=4 expressions--> Probability that their product will be of the form of x^2 - (by)^2, where b is an integer = 2/4=1/2

Hi tkorzhan1995,

Your definition of probability is mathematically correct, but you have to follow the instructions in the prompt to make sure that you are correctly applying that formula.

We're told to take the product of 2 of the four given 'expressions.' When multiplying two terms together, it does not matter which one is first - meaning that there are 6 possible pairings of the four expressions (you could list them all out or use the Combination Formula --> 4c2).

From those 6 possible pairings, there is just 1 that is a match for the 'format' defined in the prompt, thus the probability is 1/6.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at: Rich.C@empowergmat.com
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Bunuel
If two of the four expressions x+y, x+5y, x-y, and 5x-y are chosen at random, what is the probability that their product will be of the form of x^2 -(by)^2, where b is an integer?

A. 1/2
B. 1/3
C. 1/4
D. 1/5
E. 1/6

Four expressions can be paired in \(C^2_4=6\) ways (choosing 2 out of 4);

\(x^2 - (by)^2=(x-by)(x+by)\) and only one pair is making such expression: \((x+1*y)(x-1*y)\), hence probability is 1/6.

Answer: E.


Dear Bunuel, thank you for the explanation but I have one doubt-

I got answer as A - 1/3

We want : (x + by) ( x - by)

We have x + y , x - y, x + 5y , 5x - y
While (x-y)(x+y) is obvious, can we say that --> x + 5y = x - y + 6y = x-y(1-6) = x - by form?
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Bunuel
If two of the four expressions x+y, x+5y, x-y, and 5x-y are chosen at random, what is the probability that their product will be of the form of x^2 -(by)^2, where b is an integer?

A. 1/2
B. 1/3
C. 1/4
D. 1/5
E. 1/6

Four expressions can be paired in \(C^2_4=6\) ways (choosing 2 out of 4);

\(x^2 - (by)^2=(x-by)(x+by)\) and only one pair is making such expression: \((x+1*y)(x-1*y)\), hence probability is 1/6.

Answer: E.


Dear Bunuel, thank you for the explanation but I have one doubt-

I got answer as A - 1/3

We want : (x + by) ( x - by)

We have x + y , x - y, x + 5y , 5x - y
While (x-y)(x+y) is obvious, can we say that --> x + 5y = x - y + 6y = x-y(1-6) = x - by form?

If I got you right, you mean that if we write x + 5y as x - (-5)y, then it when multiplied by 5x-y will also be of the form of (x-by)(x+by). But if one of the expression is x - (-5)y, then, the product to be of the form of (x-by)(x+by), the second one must be x + (-5)y = x - 5y, while we have 5x-y.

Does this make sense?
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Hi Bunuel or any other expert.

Quote:
It should be 1/4*1/3*2, since you can choose (x+y) first and then (x-y) or vise versa.

Why did you multiply the above the 2? I think I am missing a concept here. Could you please help explain or refer to relatable links for guidance?
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Hi Bunuel or any other expert.

Quote:
It should be 1/4*1/3*2, since you can choose (x+y) first and then (x-y) or vise versa.

Why did you multiply the above the 2? I think I am missing a concept here. Could you please help explain or refer to relatable links for guidance?
See this explanation https://gmatclub.com/forum/if-two-of-th ... 92727.html
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Quote:
If two of the four expressions x+y, x+5y, x-y, and 5x-y are chosen at random, what is the probability that their product will be of the form of x^2 -(by)^2, where b is an integer?

A. 1/2
B. 1/3
C. 1/4
D. 1/5
E. 1/6

1. The six possible products are:
(x+y)(x+5y) = x^2 + 6xy +5y^2 = doesn't work
(x+y)(x-y) = x^2 - y^2 = x^2 - (1y)^2 = WORKS!
(x+y)(5x-y) = 5x^2 +4xy - y^2 = doesn't work
(x+5y)(x-y) = x^2 + 4xy - 5y^2 = doesn't work
(x+5y)(5x-y) = 5x^2 +24xy -5y^2 = doesn't work
(x-y)(5x-y) = 5x^2 -6xy + y^2 = doesn't work
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Asked: If two of the four expressions x + y, x + 5y, x - y, and 5x - y are chosen at random, what is the probability that their product will be of the form of x^2 - (by)^2, where b is an integer?

(x+y)(x-y) = xˆ2 - yˆ2 : Of the form x^2 - (by)^2, where b is an integer
(x+y)(5x-y) = 5xˆ2 +4xy - yˆ2 : NOT Of the form x^2 - (by)^2, where b is an integer
(x+y)(x+5y) = xˆ2 + 6xy + 5yˆ2: NOT Of the form x^2 - (by)^2, where b is an integer
(x-y)(5x-y) = 5xˆ2 - 6xy +yˆ2: NOT Of the form x^2 - (by)^2, where b is an integer
(x-y)(x+5y) = xˆ2 + 4xy - 5yˆ2: NOT Of the form x^2 - (by)^2, where b is an integer
(x+5y)(5x-y)= 5xˆ2 + 9xy - 5yˆ2: NOT Of the form x^2 - (by)^2, where b is an integer

The probability that their product will be of the form of x^2 - (by)^2 = 1/6

IMO E
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There are a total of 6 Ways (4C2)

The acceptable form is x^2-(by)^2

So any product that has a coefficient of x is to be rejected (except 1)

Next, No product should contain any term in the form bxy (b is an integer)

This leaves us with only one pair: x+y and x-y. Therefore the probability is 1/6

TIP: If you were to take a random guess, you can safely eliminate C and D, since there are 6 possibilities, fractions with denominators that are factors of 6 will be the only correct choices. This works if you have correctly applied the 4C2 Calculation. The random chance of getting it correct improves from 20% to 33% :)
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Hi JeffTargetTestPrep,
Thank you for the step by step explanation; the probability-based approach at the end was exactly what I was looking for!

I had a follow-up doubt: how would this approach work if the expressions were slightly modified to include more potential pairs? For example, suppose the expressions were:

x + y, x - y, x + 5y, x - 5y, and 5x - y


In this case, we now have two favorable pairs

Could you please help me understand how to break this down using the probability approach in such a scenario?
TIA!

JeffTargetTestPrep
Bunuel
spc11
If two of the four expressions x+y, x+5y, x-y, 5x-y are chosen at random, what is the probability that their product will be of the form x [square] - (by) square, where b is a constant.
A.1/2
B.1/3
C. 1/4
D.1/5
E.1/6

First, notice that we are being tested on the difference of squares. We can restate the problem as: What is the probability, when selecting two expressions at random, that the product of those expressions will create a difference of two squares? Remember, the difference of two squares can be written as follows:

a^2 - b^2 = (a + b)(a - b)

So, x^2 - (by)^2 can be written as (x + by)(x - by). Thus, we are looking for two expressions in the form of (x + by)(x - by). Although this problem is attempting to trick us with the expressions provided, the only two expressions that when multiplied together will give us a difference of squares are x + y and x - y. When we multiply x + y and x - y, the result is x^2 - y^2 or x^2 - (1y)^2.

We see that there is just one favorable product, namely (x + y)(x - y). In order to determine the probability of this event, we must determine the total number of possible products. Since we have a total of four expressions and we are selecting two of them to form a product, we have 4C2, which is calculated as follows:

4C2 = (4 x 3)/(2!) = 12/2 = 6 products

Of these 6 products, we have already determined that only one will be of the form x^2 - (by)^2. Therefore, the probability is 1/6.

The answer is E.

Note: If you don’t know how to use the combination formula, here is a method that will work equally well:

We are choosing 2 expressions from a pool of 4 possible expressions. That is, there are 2 decisions being made:

Decision 1: Choosing the first expression

Decision 2: Choosing the second expression

Four different expressions are available to be the first decision.

For the second decision, 3 remaining expressions are available because 1 expression was already chosen. We multiply these two numbers: 4 x 3 = 12.

The final step is to divide by the factorial of the number of decisions (2! = 2) because the order in which we multiply the expressions doesn’t matter (for example, (x+y)(x-y) = (x-y)(x+y)). In this case, the two expressions are only considered as one, so we need to divide 12 by 2.

12/2 = 6

Once again the answer would be E.

Alternate Solution:

One other way to solve this problem is to use probability.

Once again, we have determined that the only two expressions that when multiplied together will give us a difference of squares are x + y and x - y. If we select either of those expressions first, since there are 2 favorable expressions and 4 total expressions, there is a 2/4 = 1/2 chance that either x + y or x - y will be selected. Next, since there is 1 favorable expression left and 3 total expressions, there is a 1/3 chance that the final favorable expression will be selected.

Thus, the probability of selecting x - y and x + y is 1/2 x 1/3 = 1/6.

Answer: E
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