here is a probability question:
if two of the four expressions (x+y), (x+5y), (x-y) and (5x-y) are selected at random, what is the probability of their product being of the form x^2-(by)^2, where b is an integer? thanks in advance
These are all the different possibilities. Of these only one pair produces the required form, #2. So we need to pick two choices, of which there are only two elements we can choose. So by the counting principle we have 2 choices for the first factor, then we are left with only 1 possibility for the second choice. There are 4*3 ways to make a product of any two of the factors. So:
So we have a 1 in 6 chance of getting the required product.