2. The password consists of four different digits, the sum of which is 27. How many password options exist
Answer: 72.

Solution. Among the digits of the password, there is a 9. Otherwise, since all digits of the password are different, the maximum sum of the digits will not exceed $8+7+6+5=26$. If there are 9 and 8, then the other two digits are no more than 7, and their sum is 10. There are two possible cases: 7,3 and 6,4, resulting in the sets $9,8,7,3$ and $9,8,6,4$. If there is no digit 8, then the only set is $-9,7,6,5$. Let's count the number of possible passwords. In each set, the four digits can be permuted in $4 \cdot 3 \cdot 2 \cdot 1=24$ ways. In total, there are 24 . $3=72$ password variants.

Comment. The correct answer is obtained by the method of complete enumeration - 7 points. The correct answer is obtained by the method of incomplete enumeration, and quite obvious cases are discarded without explanation - 6 points. If the correct method of counting is used but there are errors in finding the number of variants - deduct 2 points for each error. Only established that among the digits of the password there is a 9 - 1 point. For useful ideas - 1-2 points.