How many increasing numbers are between 100 and 1000?

Answer 1

To find the number of increasing numbers between 100 and 1000, we can consider the possible combinations of digits. There are 9 choices for the first digit (1-9), 9 choices for the second digit (0-9 excluding the first digit), and 9 choices for the third digit (0-9 excluding the first two digits). Therefore, the total number of increasing numbers is 9 x 9 x 9 = 729.

Answer 2

This depends entirely on what you mean by numbers. Assuming you mean numbers in the set of naturals or integers, there are 899 numbers in the interval (100,1000).

If on the other hand you mean how many increasing rational numbers there are between 100 and 1000, the answer is, infinitely many. The same thing goes for the increasing numbers in the reals, again there are infinitely many numbers.

As a side note: due to a property of the real numbers, there are infinitely many more real numbers counting from 100 to 1000 than there are rational numbers. The proper way of saying this is that, relative to the set of real numbers, the set of rationals is a null set. In other words, even though the set of rationals has an infinite number of elements, when compared to the set of real numbers, the set of rationals has relatively no elements at all.