Factorials
The symbol n! (read “n factorial”) means the product of all integers from n down to 1.
This is the basic counting tool used later in combinations and permutations.
\[
\begin{aligned}
n! &= n\,(n-1)\,(n-2)\,\cdots\,3\cdot 2\cdot 1 \\
0! &= 1
\end{aligned}
\]
\[
\begin{aligned}
10! &= 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1 \\
&= 3628800
\end{aligned}
\]
\[
\begin{aligned}
(12-4)! &= 8! \\
&= 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1 \\
&= 40320
\end{aligned}
\]
\[
\begin{aligned}
(5-5)! &= 0! \\
&= 1
\end{aligned}
\]
Combinations
A combination counts how many different ways we can choose x elements from n distinct elements
when the order is not important. In combinations, selections are made without replacement, and we require n ≥ x.
\[
\begin{aligned}
{}_{n}C_{x} &= \frac{n!}{x!\,(n-x)!}
\end{aligned}
\]
Remember: selecting none or selecting all gives exactly one combination.
\[
\begin{aligned}
{}_{n}C_{n} &= 1 \\
{}_{n}C_{0} &= 1
\end{aligned}
\]
Example: choosing 2 flavors from 6.
\[
\begin{aligned}
{}_{6}C_{2} &= \frac{6!}{2!\,(6-2)!} \\
&= 15
\end{aligned}
\]
Permutations
A permutation counts how many different ordered selections we can make when the order matters.
Permutations are also called arrangements. Again, items are distinct and we require n ≥ x.
\[
\begin{aligned}
{}_{n}P_{x} &= \frac{n!}{(n-x)!}
\end{aligned}
\]
Example: selecting 3 office holders from 20 members (order matters).
\[
\begin{aligned}
{}_{20}P_{3} &= \frac{20!}{(20-3)!} \\
&= 6840
\end{aligned}
\]
Why permutations are larger than combinations
Every combination of x chosen elements can be arranged in x! different orders.
That is why permutations count more outcomes when x ≥ 2.
\[
\begin{aligned}
{}_{n}P_{x} &= {}_{n}C_{x}\cdot x!
\end{aligned}
\]
