Question

Two integers x and y are chosen from first 11 whole numbers Find the probability that |x - y| ​

Answer 1

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Answer 2

Answer:

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Here, x, y ε {1, 2, 3, 4, 5, 6, 7, 8, 9}.

There are (9C1) = 9 possible choices for x.

Before choosing y, there is a provision for replacement of the digit already chosen for x.

So, there are (9C1) = 9 possible choices for y as well for every choice of x.

Thus, there are (9 * 9) = 81 choices for (x, y).

Now, {(x^2) - (y^2)} = {(x + y) * (x - y)}.

So, {(x^2) - (y^2)} will be divisible by 2 if either both x and y are even numbers or both x and y are odd numbers. Even for x = y, {(x^2) - (y^2)} = 0 divisible by 0.

In the set {1, 2, 3, 4, 5, 6, 7, 8, 9}, there are four even numbers and five odd numbers.

There are (4 * 4) = 16 choices for (x, y) both being even numbers.

There are (5 * 5) = 25 choices for (x, y) both being odd numbers.

Thus, in totality, there are (16 + 25) = 41 choices for (x, y), for which {(x^2) - (y^2)} is divisible by 2.

Therefore, the probability that {(x^2) - (y^2)} is divisible by 2 is (41 / 81) ≈ 0.5061728