Bunuel wrote:
spc11 wrote:
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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