Discrete Mathematics
# Complements

In the garden of Eden, there are $8$ trees which bear fruit. There is the Tree of Knowledge of good and evil, the Tree of Life, the Abiu Tree, the Babaco Tree, the Cherimoya Tree, the Durian Tree, the Emblic Tree and the Feijoa Tree. One fine day, Adam was hungry and asked Eve to pick $4$ fruits for him. How many ways are there for Eve to present $4$ fruits, of which at least one of them is from the Tree of Life, and none of them are from the Tree of Knowledge of good and evil?

**Details and assumptions**

Eve is allowed to pick several fruits from each tree.

Since Adam only sees the final selection of fruits, the order of picking doesn't matter - presenting (Durian, Durian, Durian, Life ) is the same as presenting (Durian, Life, Durian, Durian).

Apart from the first 2 trees, the rest of the names are actual fruit trees.

In Russian folklore this was a way that a girl of marriageable age could determine her chances of getting married within a year.

She holds six long blades of grass in her hand with the ends protruding at the top and bottom. Another girl then ties the upper six ends in pairs and the lower six ends in pairs also.

If, by following this procedure the blades of grass turn into one big ring, then she will get married within a year.

So what would you say are the chances of this girl of getting married within the year?

State your answer as a decimal correct to 3 places.

A subset of $\{1,2,\ldots, 12\}$ is said to be **selfish** if it contains its size as an element. How many subsets of $\{1,2,\ldots 12\}$ have the property that both the subset and its complement is selfish?

**Details and assumptions**

As an explicit example, the set $\{3,6,9\}$ is selfish because it contains 3 elements, and contains the element 3.

A game costs $150 to play. In this game, you roll a fair six-sided die repeatedly until each of all the six numbers has been rolled at least once. You are then paid 10 times the number of rolls you made.

For example, if the rolls were 3, 5, 4, 3, 2, 5, 1, 4, 1, 3, 6, then you would get $(10)(11) = 110$ dollars.

Including the price to play, what is your expected value in this game?

Bob has several pennant flags, each of which is either red, white or blue. To celebrate his national pride in the upcoming Olympics, he wants to string together 9 pennant flags, such that each of the colors is represented at least once, and no two consecutive flags have the same color.

How many different strings of 9 flags can Bob create?

**Note**: The string is an arrangement of 9 flags from left to right. 2 strings are different if the arrangements of the colors of those 9 flags are different.