1. ## Factorial story sums

A Quidditch team consists of seven players: three Chasers, two Beaters, one Keeper and a Seeker.

a) The Puddlemere United club has thirty-two active players. Assuming that every Puddlemere player is fanatical enough to play in any position, how many different teams can the club field?

b) The Chudley Cannons have 23 players under contract. Three of these players specialise at playing as Keepers, while 8 prefer to be Beavers. The remaining 12 players are happy to play either as Chasers or Seekers. How many different teams can the Chudley Cannons form?

I really get confused when it comes to really long factorial questions and have no idea where to start

2. Hello, SyNtHeSiS!

You can "talk" your way through these . . .

A Quidditch team consists of seven players: .$\displaystyle \begin{array}{c}\text{3 Chasers} \\ \text{2 Beaters} \\ \text{1 Keeper} \\ \text{1 Seeker}\end{array}$

a) The Puddlemere United club has 32 active players.
Assuming that every Puddlemere player can play in any position,
how many different teams can the club field?

From the 32 players, choose 3 Chasers: .$\displaystyle {32\choose3}\:=\:4960$ ways.

From the remaining 29 players, choose 2 Beaters: .$\displaystyle {29\choose2}\:=\:406$ ways.

From the remaining 27 players, choose 1 Keeper: .$\displaystyle {27\choose1} \:=\:27$ ways.

From the remaining 26 players, choose 1 Seeker: .$\displaystyle {26\choose1} \:=\:26$ ways.

Therefore, there are: .$\displaystyle 4960 \times 406 \times 27 \times 26 \;=\;1,\!413,\!659,\!520$ possible teams.

b) The Chudley Cannons have 23 players under contract.
Three of these players specialise at playing as Keepers, while 8 prefer to be Beaters.
The remaining 12 players are happy to play either as Chasers or Seekers.
How many different teams can the Chudley Cannons form?
Basically, it's the same procedure.

From the 3 available Keepers, choose 1: .$\displaystyle {3\choose1} \:=\:3$ ways.

From the available 8 Beaters, choose 2: .$\displaystyle {8\choose2} \:=\:28$ ways.

From the remaining 12 players, choose 3 Chasers: .$\displaystyle {12\choose3} \:=\:220$ ways.

From the remaining 9 players, choose 1 Seeker: .$\displaystyle {9\choose1} \:=\:9$ ways.

Therefore, there are: .$\displaystyle 3 \times 28 \times 220 \times 9 \:=\:166,\!320$ possible teams.

3. How do you know that the Quidditch team applies to the question cause they were talking about Puddlemere United club and howcome you didnt say n = 23 in b) as there 23 players?