Problems for choosing a subset of a set

Problem3. We call a natural number ``odd-looking'' if all its digits are odd. How many four-digit odd-looking numbers there are?
Solution. It is obvious that there are 5 one-digit odd-looking numbers. We can add another odd digit to the right of any odd-looking one digit number in 5 ways. Thus, we have $5\cdot5=25$ two-digit odd-looking numbers. Similarly, we get $5\cdot5\cdot5=125$ three-digit odd-looking numbers and $5\cdot5\cdot5\cdot5=5^4=625$four-digit odd-looking numbers.

Methodological Remark. In the last problem the answer has the form mⁿ. Usually, an answer of this type results from problems where we can place an element of some given set in each of n different positions, In such problems the students may encounter difficulty distinguishing the two numbers m and n, therefore confusing the base and exponent.

Problem4. How many ways are there to fill a Special Sport Lotto card? In this lotto you must predict the results of 13 hockey games, indicating either a victory for one of two teams or a draw.
Answer. 3¹³.

Problem5. The Hermetian alphabet consists of three only letters A, B, and C. A word in this language is an arbitrary sequence of no more than 4 letters. How many words does the Hermetian language contain?
Hint. Calculate separately the numbers of one-letter, two-letter, three-letter, and four-letter words.

Answer. 3+3²+3³+3⁴.

Problem6. How many ways are to put one white and one black rooks on a chessboard so that they do not attack each other.
Solution. The white rook can be placed on any of the 64 squares. No matter where it stands, it attacks exactly 15 squares (including the one it stands on). This leaves 49 squares where the black rook can be placed. Hence there are exactly $64\times49=3136$ different ways.