If a task can be performed in $n_1$ ways or in $n_2$ ways, where none of the $n_1$ ways is the same as any of the $n_2$ ways (mutually exclusive), then the task can be performed in $n_1 + n_2$ ways. More generally, if there are $k$ mutually exclusive ways to perform a task with $n_1, n_2, \ldots, n_k$ possibilities respectively, then the total number of ways is $n_1 + n_2 + \cdots + n_k$

If a procedure consists of two consecutive tasks, where the first task can be performed in $n_1$ ways and for each of these ways the second task can be performed in $n_2$ ways, then the entire procedure can be performed in $n_1 \times n_2$ ways. More generally, if a procedure consists of $k$ tasks performed sequentially with $n_1, n_2, \ldots, n_k$ ways respectively, then the total number of ways is $n_1 \times n_2 \times \cdots \times n_k$

A permutation of $n$ distinct objects taken $r$ at a time is an ordered arrangement of $r$ objects from the set of $n$ objects. The number of such permutations is denoted $P(n,r)$ or $_nP_r$ and is given by $P(n,r) = \frac{n!}{(n-r)!} = n(n-1)(n-2)\cdots(n-r+1)$

When $r = n$, we have $P(n,n) = n!$, the number of ways to arrange all $n$ objects.

A combination of $n$ distinct objects taken $r$ at a time is a selection of $r$ objects from the set of $n$ objects where order does not matter. The number of such combinations is denoted $C(n,r)$, $_nC_r$, or $\binom{n}{r}$ (binomial coefficient) and is given by $\binom{n}{r} = \frac{n!}{r!(n-r)!}$