Question

How many 3-digit numbers contain exactly one 1?

Answer 1

Given a 3-digit number containing exactly one "1". 
(ASSUME base-10 numbers, so digits are from 0-9)

Number contains exactly one "1", therefore the two other digits are limited to 2,3,4,....9,0.
If the "1" is in the leftmost position, there are 9*9=81 ways to arrange the second and third positions.
If the "1" is in the second or third position, then there are 8 choices for the leftmost digit (2-9), and 9 choices for the other digit for a total of 2*8*9=144 arrangements.
So the total number of 3-digit numbers containing exactly one "1" is 81+144=225.

Answer 2

I believe the answer would be 17, because 100, 210, 201, 301, 310, 401, 410, 501, 510, 601, 610, 701, 710, 801, 810, 901, 910