A game uses a deck of $n$ different cards, where $n$ is an integer and $n \geq 6.$ The number of possible sets of $6$ cards that can be drawn from the deck is $6$ times the number of possible sets of 3 cards that can be drawn. Find $n$. (AIME II 2005)

6 people are separated into 3 rooms, $A,B$ and $C$. How many ways can this be done if none of the rooms are empty?

2 Spades $\spadesuit$ and 2 Hearts $\heartsuit$ are taken from a deck of cards and arranged in a line. How many arrangements are there if there is at least one face card? *a face card is a either a Jack, Queen or King

There are 4 girls and 3 boys. When arranged, a boy cannot be next to another boy on either side. How many different ways can you arrange the 7 children:
a) in a line?
b) in a circle?

How many different ways can you arrange 3 letters in a row taken from the letters A, A, A, B, B and C?

There are 5 fiction books and 4 non-fiction books in a basket. How many different ways can 4 students choose books? It is also possible for a student to not choose a book at all.