Key Points

If an event can occur in $m$ different ways, and another event can occur in $n$ different ways, then the total number of occurrences of the events in the given order is $m \times n$. This is also known as the multiplication principle.

A permutation is an arrangement of a number of objects in a definite order. In permutations, the order of the objects is important.

The notation $n!$ (n factorial) represents the product of the first $n$ natural numbers. It is defined as $n! = n \times (n-1) \times ... \times 2 \times 1$. By definition, $0! = 1$.

The number of permutations of $n$ different objects taken $r$ at a time, where repetition is not allowed, is denoted by ${}^n P_r$ and calculated as ${}^n P_r = \frac{n!}{(n-r)!}$, where $0 \leq r \leq n$.

The number of permutations of $n$ different objects taken $r$ at a time, where repetition is allowed, is $n^r$.

The number of permutations of $n$ objects, where $p_1$ objects are of one kind, $p_2$ are of a second kind, ..., and $p_k$ are of a $k^{th}$ kind, is given by the formula $\frac{n!}{p_1! p_2! ... p_k!}$.

A combination is a selection of a number of objects where the order of selection does not matter. It is about choosing a group, not arranging it.

The number of combinations of $n$ different objects taken $r$ at a time is denoted by ${}^n C_r$ and calculated as ${}^n C_r = \frac{n!}{r!(n-r)!}$, where $0 \leq r \leq n$.

The number of permutations is the number of combinations multiplied by the number of ways to arrange the selected items. The formula is ${}^n P_r = {}^n C_r \times r!$.

Selecting $r$ objects from $n$ is the same as rejecting $(n-r)$ objects. This is represented by the formula ${}^n C_r = {}^n C_{n-r}$.

If ${}^n C_a = {}^n C_b$ for distinct $a$ and $b$, it implies that $a + b = n$.

This rule relates adjacent values in Pascal's triangle and is given by the formula ${}^n C_r + {}^n C_{r-1} = {}^{n+1} C_r$.