How many different numbers can be formed by using only the digits 1 and 3 which are smaller than 3000000?

For any number which has less than 7 digits, each place in the number can be filled in two ways [either 1 or 3].

Total number of 1-digit numbers = 2

Total number of 2-digit numbers = 2 x 2 = 4

Total number of 3-digit numbers = 2 x 2 x 2 = 8

Total number of 4-digit numbers = 2 x 2 x 2 x 2 = 16

Total number of 5-digit numbers = 2 x 2 x 2 x 2 x 2 = 32

Total number of 6-digit numbers = 2 x 2 x 2 x 2 x 2 x 2 = 64

For 7-digit numbers, we need the number to be less than 3000000, thus the first digit can only be, and the rest digits can be filled in 2 ways. Thus, 

Total number of 7-digit numbers = 1 x 2 x 2 x 2 x 2 x 2 x 2 = 64

Total numbers = 2 + 4 + 8 + 16 + 32 + 64 + 64 = 190