How many different numbers which are smaller than 2108 can b

How many different numbers which are smaller than 2.10^8 can be formed using the digits 1 and 2?

Solution

2*10⁸ has 9 digits and the number is = 200,000,000 (which is basically 200 million)
Now we have to form a number that is lower than 200 million by only using the digits 1 and 2

So there are 9 places (digits) each of which can be filled with either a \"1\" or a \"2\";
However if we fill the first digit with a 2 then the second cannot be a \"1\" or a \"2\" as it will make the number 21x million or 22y million which are greater than 200 million
However if the first digit is \"1\" then the remaining 8 digits can be filled in any manner with any sequence of \"1\"s or \"2\"s
Hence the first digit can be filled in only one way (with the digit \"1\") and each of the remaining 8 digits can be filled in 2 ways each (with a \"1\" or a \"2\")

Hence total numebr of numbers = 1* 2* 2* 2* 2*2*2*2*2= 2⁸
There are 2⁸numbers below 200 million that can be formed with only the digits 1 and 2