Any help you guys could give me to figure these out would be great, I've been thinking about them for hours, I'll need to be able to do this type of stuff on a test and Im completely stuck. I think maybe its the variables taking place of numbers that's messing me up, and maybe because I've done too much math tonight. I don't really need full solutions, just some hints of how to do these would be great. Thanks a lot :)
1. A gardener has 12 tulip bulbs to plant in a box (a 4 columnsX3 rows grid). There are 4 white, 4 red, and 4 yellow bulbs. and exactly 1 bulb is planted in each small square.
a) How many different colour patterns are possible? I think this one's
b) How many of these patterns have each row entirely the same color?
c) How many of these patterns have the four corner squares the same color?
2. A subset of size r is formed from a set of size n that includes the symbols A and B. How many of these contain A or B?
3. Determine n(AUB) such that
A=set of sequences of length 3 made from the letters GEOMETRY
B=set of sequences of length 3 made from the letters DISCRETE
4. Given a set of n symbols, how many sequences of length 2r (where 2r<n) can be formed such that every symbol used is included in the sequence twice?
5. How many binary sequences of length 2n have the same number of 1s in the first and second half of the sequence?