Fundamental Counting Principle

The Fundamental Counting Principle states that if there are a ways of doing something and b ways of doing another thing, then there are a x b ways of performing both actions.
This can be extended to more than two activities. If there are a ways of doing one thing, b ways of doing a second thing, c ways of doing a third thing, etc., then there are a x b x c x … ways of performing all actions.

Permutations

Permutations are arrangements of objects where the order is important.
The number of permutations of n distinct objects is n! (n factorial).
If we are considering permutations of r objects taken from a group of n distinct objects, the number of permutations is nPr = n! / (n-r)!.
If there are duplicate items in the group from which we are choosing, use the formula for permutations of a multiset: n! / (n1! x n2! x…). n is the total number of items, n1 is the number of the first type of item, n2 is the number of the second type of item, etc.

Combinations are selections of objects where the order does not matter.
The number of ways to choose r objects from a group of n distinct objects is given by nCr = n! / [r!(n-r)!].
If there are duplicate items in the group from which we are choosing, use the formula nHr = (n+r-1)Cr. n is the number of types of items, r is the number of items to choose.

Binomial Coefficients are used when we are choosing items with replacement.
The binomial coefficient can be computed using the formula: C(n, k) = n! / [k!(n-k)!].
Binomial coefficients have numerous applications in probability theory and are used extensively in the binomial theorem.

The Pigeonhole Principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item.
This principle is a simple yet powerful tool used to prove the existence of certain numbers or configurations.