Hi,
Need help answering the question in the attached screen shot.
Binara
Prob of product
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Important concept tested hereIf 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 x² - (by)², where b is an integer?
A) 1/2
B) 1/3
C) 1/4
D) 1/5
E) 1/6
First recognize that x² - (by)² is a DIFFERENCE OF SQUARES.
Here are some examples of differences of squares:
x² - 25y²
4x² - 9y²
49m² - 100k²
In general, we can factor differences of squares as follows:
a² - b² = (a-b)(a+b)
So . . .
x² - 25y² = (x+5y)(x-5y)
4x² - 9y² = (2x+3y)(2x-3y)
49m² - 100k² = (7m+10k)(7m-10k)
--------------------------
From the 4 expressions (x+y, x+5y ,x-y and 5x-y), only one pair (x+y and x-y) will result in a difference of squares when multiplied.
So, the question now becomes:
If 2 expressions are randomly selected from the 4 expressions, what is the probability that x+y and x-y are both selected?
P(both selected) = [# of outcomes in which x+y and x-y 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, we have a free video on calculating combinations (like 4C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789
# of outcomes in which x+y and x-y are both selected
There is only 1 way to select both x+y and x-y
So, P(both selected) = 1/6 = E
Cheers,
Brent
GMAT/MBA Expert
- Brent@GMATPrepNow
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- Joined: Mon Dec 08, 2008 6:26 pm
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Alternate solution: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 x² - (by)², where b is an integer?
A) 1/2
B) 1/3
C) 1/4
D) 1/5
E) 1/6
Once we recognize (from my last post) that we must find P(x+y and x-y are both selected), we can write:
P(x+y and x-y are both selected) = P(one of the two expressions is selected on 1st pick AND the other expression is selected on 2nd pick)
= P(one of the two expressions is selected on 1st pick) x P(the other expression is selected on 2nd pick)
= 2/4 x 1/3
= [spoiler]1/6[/spoiler]
= E
Cheers,
Brent