Some problems from Estonian Mathematical Olympiad: 2000

Problem 1: How many positive integers less than 20002001 and not containing digits other than 0 and 2 are there?

Solution 1: The set of integers under consideration consists of all integers with up to 7 digits containing only digits 0 and 2, all 8 digit integers of the form 20000***, and the integer 20002000. There are $2^{k-1}$ integers with exactly k digits 0 and 2, and $2^{3}$ integers of the form 20000***. So, the required number of integers is $(2^{0}+2^{1}+2^{2}+ \ldots + 2^{6})+8+1=(2^{7}-1)+9=136$

Problem 2: Find the last two digits of the number: $1! + 2! + 3! +4! + \ldots +2000!$

Solution 2: This is a classic type of question: the product $1.2 \ldots 10$ has 2, 5, and 10 as factors, therefore being divisible by 100. Hence, the last two digits of n! are zeros for any $n \geq 10$ and it suffices to find the last two digits of $1!+2! + 3! + \ldots + 9!$ . The last two digits of the summands are 01, 02, 06, 24, 20, 20, 10, 20 and 80, yielding 13 as the answer.