Question

We call a number special if every digit in the number either is a 1 or borders a 1. For example, 11111, 13, 141, 1441, 515151, and 101 are all special, but 10001, 222, 122, and 1333 are not special. How many positive 3-digit numbers are special?

Answer 1

Let us say that:

T = numbers from 2-9 
Z = numbers from 0-9 
X = all except 1 (0, 2-9)

If a number begins with 1: 
1 _ _ 
Then we can have 
1 1 Z (10 numbers) 
1 X 1 (9 numbers) 
So a total of 19. 

Now if a number starts with 2-9 
T _ _ 
The second digit must be 1 

T 1 _
but the third can be anything so
T 1 Z = 8 * 1 * 10 = 90 numbers 

Total = 99