-> (Multiplication Principle) If there are n choices for the first step of a two step process and m choices for the second step, the number of ways of doing the two step process is nm.
-> The number of arrangements of n objects is n!-> The number of arrangements of r out of n objects is nPr = n!/(n-r)!
-> The number of arrangements of n objects in a circle is (n-1)!
-> The number of arrangements of n objects on a key ring is (n-1)!/2
-> The number of arrangements of n objects with r1 of type 1, r2 of type 2, ..., ri of type i is n!/(r1!r2!...ri!)
-> The number of ways of choosing n out of r objects is nCr = n!/((n-r)! r!)
-> The number of distributions of n distinct objects in k distinct boxes is kn.
-> The number of ways of distributing n identical objects in k distinct boxes is (n+k-1)Cn.
-> The sum of the coefficients of the binomial expression (x+y)n is 2n.
-> To find the sum of the coefficients of a power of any polynomial, replace the variables by 1.
-> When solving an equation for integer solutions, it is important to look for factoring. Important factoring forms:
5. If n is odd, an+bn=(a+b)(an-1-an-2b+an-3b2-...-abn-2+bn-1) (alternate signs)