Like Guest said, there are 999 - 100 + 1 = 900 numbers between 100 and 999 inclusive.

Now, think about it. The hundreds. Let's tackle 100. 107, 117, 127, 137, 147, 157, 167, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 187, and 197. How many numbers is that? Counting, it is a list of 19 numbers. 19 x 9 = 171. Why is it times 9, not 10? Well, we gotta skip the 700s because there is a 7 in all of them. In 700 - 799, there are 100 numbers with a 7 in it. 100 + 171 = 271. Therefore, 900 - 271 = 629 numbers 100 - 999 which can be written without 7's.