These are all basic permutation problems. To figure out how many combinations can be made, you would multiply the amount of choices for the first place by the amount of options for the second place, then multiply that by the amount of choices for the third place, and so on.
But be careful -- sometimes using one for the first place means that you can't use it again for the second place, as in the first three options.
You can also do this visually, by making a tree diagram: draw all your options for the first place. Then by each one draw what can go after it. Etc.
For example, let's do #1:
The scrabble tray has seven letters, and each can be used only once. You want a four-letter arrangement, which means there are four positions to be filled. So you would do 7x6x5x4, because there are seven choices for the first letter, and once one of those is used you only have 6 options for the second, and so on. (in contrast to the license plate problem, in which you have the same amount of choices for each letter space because you can use a letter more than once. So that would be 26x26 for the letters, then x9x9x9 for the three digits)
The visual method would look like this:
...and so on. This could take forever, but it helps to do it once or twice so the math version makes sense.