In a competition there are $18$ competitors. Answer the following:
A) During the first day they're competing in three-man teams (a total of $6$ teams). How many ways are there to select the teams?
B) If the main sponsor of the event demands that the four best-ranked players mustn't play in the same groups, how many ways can you then select the teams?
I've tried the following:
A) There are $(18\cdot17\cdot16 + 15\cdot14\cdot13 + 12\cdot11\cdot10 +...+ 3\cdot2\cdot1)$ ways to select the teams, and if the order of the groups mattered we would just multiply that with $6!$
However, this gives me approximately $700000$ ways, whereas the correct number of ways according to the book is $190000000$
B) The four best players must each be in a different group. Therefore we "lock" their positions. $(1\cdot14\cdot13 + 1\cdot12\cdot11 + 1\cdot10\cdot9 + 1\cdot8\cdot7 + 6\cdot5\cdot4 + 3\cdot2\cdot1)$ gives us the ways to select the teams, and if the order of the groups matters, we multiply that by $6!.$
Once again, this value is far too small according to the book. Either my calculations are way off or the book is wrong.