Summary

• The fundamental principle of counting (multiplication rule): if one event can occur in $m$ ways and another independent event in $n$ ways, the two together can occur in $m \times n$ ways. The addition rule applies when events are mutually exclusive: $m + n$ ways.

• A permutation is an arrangement of objects in a definite order. The number of permutations of $n$ distinct objects taken $r$ at a time is:

Special case: $^nP_n = n!$ (arranging all $n$ objects).

• When objects are repeated (not all distinct), the number of distinct permutations of $n$ objects where there are $p$ of one kind, $q$ of another, etc., is:

• A combination is a selection of objects without regard to order. The number of combinations of $n$ distinct objects taken $r$ at a time is:

• Key relation: $^nC_r = ^nC_{n-r}$. Also, $^nC_0 = ^nC_n = 1$, and $^nC_1 = n$. …