Your friend Carol loves to prepare gifts of marbles in bags for children. On Monday, she has five red marbles, six blue marbles and seven white ones.

*Warm-Up:*

How many different bags can she pack so each bag will have a different collection of marbles? Two bags are different if for at least one color, one bag has a different number of marbles from the other. For example, a bag with two red marbles and one blue is different from a bag having one red and one blue or a bag having one red and two blue.

*Solution to Warm-Up:*

For brevity, the colors will be represented by their first letters: R (red), B (blue), and W (white). The commas indicate distinct bags. Carol can pack 10 different bags:

R, B, W, RR, BB, WW, RB, BW, RW, BWW.

That was just the warm-up. Now let's try something more challenging.

**Problems:**

**1.** On Tuesday, she again starts with five red marbles, six blue marbles and seven white ones. In order to give the children more opportunities to trade, she wants to pack the bags so that each bag differs by at least two "moves" from any other bag. A move consists of inserting a marble into a bag or removing a marble from a bag. So for example, R and RR differ by only one move (insert R). On the other hand, RW and RB differ by two (remove W and insert B). How many different bags of marbles can Carol create that differ by at least two moves?

**2.** On Wednesday, Carol is feeling physically better, but she is troubled by the fact that she didn't select the marbles herself. She knows she has 18 of them in total. She also knows that there is at least one red, at least one blue, and at least one white. Knowing only this much and before seeing the marbles, Carol receives a phone call. "Can you guarantee to be able to give each child a bag with a different collection of marbles (i.e. at least one move apart) if there are eight children at a party? If not, then can you make this guarantee if there are seven children? If so, then how about nine?" How should Carol answer?

**3.** On Thursday, she is in the same situation as Wednesday with respect to her knowledge of her marble supply (she knows only that she has 18 marbles and at least one of each color) and in the same situation as Tuesday as far as her wanting the children to trade. So she wants each gift to differ from every other by at least two moves. How many children can she guarantee to prepare bags for in that case?

**4.** On Friday, her friend Diane packs the bags. Diane assures Carol (i) that every bag contains a different (by at least one move) collection of marbles, (ii) that there were 18 marbles to start with, (iii) that there are more whites than blues and more blues than reds, (iv) and that the maximum number of different bags she could pack was seven. Assuming that Diane is an extremely capable packer, what is the maximum number of red marbles that there could have been?