aditiniyer wrote:If 2 of the 4 expressions x+y, x+5y,x-y & 5x-y are chosen at random, what is the prob that their product will be of the form x^2-by^2 where b is an integer.
Ans 1/6, I got 1/4
Many nice solutions are provided by experts. Here is my take on this question.
We have to choose two expressions out of four: (x+y), (x+5y), (x-y) & (5x-y) such that the product of them is in the form of (x^2-by^2), where b is an integer.
We see that in the desired quadratic expression (x^2-by^2), there is no term of 'xy', thus, we must choose the TWO terms such that we avoid 'xy' terms when they are multiplied.
Let's analyses possible pairs of two terms in (x+y), (x+5y), (x-y) & (5x-y).
1. (x+y) and (x+5y): It will have a 'xy' term when multiplied, so ignore it.
2. (x+y) and (x-y) = x^ - y^2: This in the form of x^2-by^2. Here b = 1. We need this!
3. (x+y) and (5x-y): It will have a 'xy' term when multiplied, so ignore it.
4. (x+5y) and (x-y): It will have a 'xy' term when multiplied, so ignore it.
5. (x+5y) and (5x-y): It will have a 'xy' term when multiplied, so ignore it.
6. (x-y) and (5x-y): It will have a 'xy' term when multiplied, so ignore it.
Out of six outcomes, the desired one is only one outcome: #2.
Thus, the probability = [spoiler]1/6[/spoiler]
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Hope this helps!
-Jay
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