# If you have 20 dice (with 6 faces each), how many different numbers can you create putting the dice in one line and interpreting each face as a digit?

Jan 14, 2016

${6}^{20}$
(a big number: according to my calculator in the order of $3.65616 \times {10}^{15}$)

#### Explanation:

There are 6 different possible first die values; each of these gives a different number.
So the number of different numbers you can get with 20 dice is:
$\textcolor{w h i t e}{\text{XXX}} 6 \times$the number of different numbers you can get with 19 dice

Similarly the number of different numbers you can get with 19 dice is:
$\textcolor{w h i t e}{\text{XXX}} 6 \times$ number of different numbers you can get with 18 dice
or
The number of different numbers you can get with 20 dice is
$\textcolor{w h i t e}{\text{XXX}} 6 \times 6 \times$ number of different numbers you can get with 18 dice.

Continuing this pattern, we see that
the number of different numbers you can get with 20 dice is
$\textcolor{w h i t e}{\text{XXX}} 6 \times 6 \times 6 \times \ldots$ for a total of 20 times
or
$\textcolor{w h i t e}{\text{XXX}} {6}^{20}$