Factorial, permutations, combinations

Permutations:

• How many ways can Alice, Ben, Chaz, Dave and Elli arrange themselves to sit on a long chair?
  
  
  
  
  
  
  
  

• How many ways can they arrange themselves around a circular table?
  
  
  

Comibination:
How many ways of choosing 3 people out of 5 people are there?

We use ${n \choose k}$ (pronounciation is ``n choose k'') to denote how many ways of chooing k items out of n items.

Note the distinction between combinations and permutations: A permutation is an arrangement or an ordering - there are 6 ways to order (permute) 2 out of 3 objects: ABC $\Rightarrow$ AB, BA, AC, CA, BC, CB. A combination pays no attention to ordering so there are only 3 ways to combine 2 out of 3 objects: ABC $\Rightarrow$ AB, AC, BC.)