The product rule

If there is a sequence of $k$ tasks such that:

• there are $n_1$ ways to do the first task
• $n_2$ to do the second
• $n_k$ to do the $k$th task ? Then, there are $n_1\cdot n_2\cdot ...\cdot n_k$ ways to do the whole sequence of $k$ tasks

$$|A_1\times A_2\times ...\times A_k|=|A_1|\cdot |A_2|\cdot ...\cdot |A_k|$$

title: Example 1
If each number plate contains a sequence of three letters followed by three digits (and no such sequence is prohibited), then the number of available different number plates is:
 
$$
	26 \cdot 26 \cdot 26 \cdot 10 \cdot 10 \cdot 10 = 17, 576, 000
$$

Usage with functions

If $|A|=n$ and $|B|=m$ then the number of different $f:A\to B$ functions is: ?

$$\stackrel{n}{\overbrace{m\cdot m\cdot ...\cdot m}}=m^n$$

title: Example 2 (with functions)
Let $A = \{ a,b,c \}, B = \{ d,e \} \rightsquigarrow n = 3, m = 2$.
- There are $2$ ways to choose a value for $a$.
Then there are $2$ ways to choose a value for $b$ and then $c$.
 
Hence $2^3 = 8$.