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 of x2 - (by) 2, where b is an integer?

1/6

Step-by-step explanation:

The product will only be of the required form if the two chosen factors are of the form (x + by) and (x - by).

For the factor (x+y), there is also a factor (x-y) on the list. For the factor (x+5y), there is no factor (x-5y) available.

So, of the (4·3) / 2 = 6 possible ways to choose two expressions from the four given expressions, only one pair of the 6 will have a product of the required form.

The probability is 1/6.