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Bunuel
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If two of the four expressions a+b, 3a+b, a-b, and a-3b are chosen at 31 Jan 2022, 06:45
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If two of the four expressions a+b, 3a+b, a-b, and a-3b are chosen at random, what is the probability that their product will be of the form of a^2 - kb^2, where k is a positive integer?

A. 1/8
B. 1/6
C. 1/4
D. 1/3
E. 1/2


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If two of the four expressions a+b, 3a+b, a-b, and a-3b are chosen at 31 Jan 2022, 13:52
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Hi Bunuel,

I think you meant to write "...of the form of a^2 - kb^2..." (rather than a^2 + kb^2)
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If two of the four expressions a+b, 3a+b, a-b, and a-3b are chosen at 01 Feb 2022, 01:09
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BrentGMATPrepNow
Hi Bunuel,

I think you meant to write "...of the form of a^2 - kb^2..." (rather than a^2 + kb^2)
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Fixed the typo. Thank you Brent!
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If two of the four expressions a+b, 3a+b, a-b, and a-3b are chosen at 01 Feb 2022, 08:19
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Bunuel
If two of the four expressions a+b, 3a+b, a-b, and a-3b are chosen at random, what is the probability that their product will be of the form of a^2 - kb^2, where k is a positive integer?

A. 1/8
B. 1/6
C. 1/4
D. 1/3
E. 1/2
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Okay, first recognize that a² - kb² closely resembles a DIFFERENCE OF SQUARES (since there are no ab terms).
For example, if k = 1, we get a² - kb² = a² - b² = (a + b)(a - b) [notice that the ab terms cancel out when we expand (a + b)(a - b)]

So, from the 4 expressions (a+b, 3a+b, a-b, and a-3b), only one pair (a+b and a-b) will result in an expression in the form a² - kb².

So, the question becomes: If 2 expressions are randomly selected from the 4 expressions, what is the probability that a+b and a-b are both selected?

P(both selected) = [# of outcomes in which a+b and a-b are both selected]/[total # of outcomes]

As always, we'll begin with the denominator.

total # of outcomes
There are 4 expressions, and we must select 2 of them.
Since the order of the selected expressions does not matter, we can use combinations to answer this.
We can select 2 expressions from 4 expressions in 4C2 ways (= 6 ways)

If anyone is interested, I have added a video (below) on calculating combinations (like 4C2) in your head

# of outcomes in which a+b and a-b are both selected
There is only 1 way to select both a+b and a-b

So, P(both selected) = 1/6

Answer: B

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If two of the four expressions a+b, 3a+b, a-b, and a-3b are chosen at 22 Jun 2025, 11:18
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