varun289 wrote:260. 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 x^2-(by)^2, where b is an integer?
A. 1/2
B. 1/3
C. 1/4
D. 1/5
E. 1/6
Solution:
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!/[(4-2)! x 2!] = 4!/(2! x 2!) = (4x3x2x1)/(2x1x2x1) = 24/4 = 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 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, we had 2 decisions so we divide by divide by 2!.
(4x3)/2! = 12/2 = 6
Once again the answer would be E.
