Let us examine . Draw a Venn diagram with a square around it symbolizing the outside of . Now there are possible locations to put a number if you drew out the diagram. We need to put these numbers into these locations in such a way that i.e. so that no two circles intersect. That means a number can go into four remaining locations: either only into or or or completely outside the circles. Therefore we are placing 'objects' into 'boxes' and the total number is

If general we would have to draw circles which partition the plane into regions. The total number of places to put the numbers is . The number of ways of doing this is

No this is not the same as problem 1. This is included in problem 1 therefore the number would be smaller.

Again draw the Venn diagrams for circles. This time the numbers are being places anywhere except into the middle of where all three circles intersect. That leaves us with boxes where we place numbers. The total number is therefore,

Try doing the others ones.

(This is how I get out of doing all the problems when I get lazy).