In Probability and Statistics, it is an important first step to determine all the possible outcomes of an experiment using Counting and Combinatorial Number Theory methods

• direct enumeration
• permutations
• combinations

combination formula/binomial coefficient

• gives us the number of ways of choosing x items (w/o replacement) from n distinct items if order of selection is not important
• choose(n,x) on R
• $$\left(\begin{array}{c}n\\ x\end{array}\right)=\frac{n!}{x!(n-x)!}$$
• example
  • lotto 6/49, buyers pick 6 numbers from 1-49
  • how many different tickets are possible
  • $$\left(\begin{array}{c}49\\ 6\end{array}\right)=\frac{49!}{6!(49-6)!}$$