Q6: How many squares of each color are there? Pretend both checkers will go on a black square; in how many ways can you choose a square for the first checker? The second? Use this to figure out in how many ways you can place both checkers on a black square. Now do the same if you want to place both on a red square.

Q7: If all letters were different, you would have $7!$ ways to arrange them, since there are $7$ letters. There are, however, some repetitions as you could swap the $N$'s and still have the same word, for example $BAN_1AN_2AS$ yields the same word as $BAN_2AN_1AS$. Thus we need to divide $7!$ by the number of ways we can swap repeated letters. In how many different ways can you swap the $N$'s? The $A$'s?

Q9: Pretend $ART$ is one letter, say $X$. So you want to figure out in how many ways you can rearrange the letters in the word $XEEEM$. Doing this is similar to Q7.

Q17: Since we are choosing points from a square of length $1$, the sample space has area $1^2 = 1$. A randomly selected point lies within $\dfrac{1}{2}$ units of the center of the square if it lies inside a circle with the same center as the square and radius $\dfrac{1}{2}$. What is the area of this circle? You can then use this areas to find the desired probability. 

The book contains 9 chapters, one for each of the first 9 lectures. The last lecture is a review of  past contest problems and is not included in the book.