Question

A ship is carrying 30 travelers from various great houses on a long sea voyage from Braavos to King’s Landing where they will participate in the Game of Thrones:

• 5 are from House Targaryen(TAR), and 5 from House Lannister (LAN)
• 4 are from House Stark (STA) and 4 from House Tyrell (TYR)
• 3 are from House Baratheon (BAR) and 3 from House Martell (MAR)
• 1 from each of the following houses: House Arryn (ARR), House Tully (TUL), House Greyjoy (GRE), House Bolton (BOL), House Frey (FRE), House Mormont(MOR)

You are now asked to evaluate probabilities relating to this voyage.

The following 9 competitors end up leaving the ship and participating in the Game of Thrones:
3 from TAR, 2 from LAN, 2 from STA, 1 from GRE' and 1 from MAR.

At the end of the Game of Thrones,only 3 of the 9 competitors will win titles:

• 1 will win the Iron Throne (I),
• 1 will end up being the Hand of the King (H)
• 1 will be the King of the North (N)

Nobody can win more than one title.

1. In how many different ways can these 3 titles be distributed among the 9 competitors?
2. In how many different ways can these 3 titles be distributed among the 5 participating houses? (in other words we are asking you to figure out how many possible combinations of titles by houses there can be.) Your answer should not be derived by listing all the possibilities one by one. Instead you should derive your answer by reasoning with known formulas for permutations and combinations.
3. Based solely on the number of competitors per house and not on their ability to wield a sword, axe, or their ability to devise a cunning plan, what is the probability that GRE will win at least one title?
4. Based solely on the number of competitors per house and not on their ability to wield a sword, axe, or their ability to devise a cunning plan, what is the probability that LAN will win at least one title?
5. Based solely on the number of competitors per house and not on their ability to wield a sword, axe, or their ability to devise a cunning plan, what is the probability that TAR will win at least one title?

Answer 1