#### Answer

About a month, (non-stop 24hrs/day)

#### Work Step by Step

The number of ways in which a series of successive things can occur is found by multiplying the number of ways in which each thing can occur.
------------
Applying the Principle on
12 different buns,
30 sauces,
4 types of lettuce,
and 3 types of tomatoes.
The total number of possible hamburgers is
$12\times 30\times 4\times 3=4320$ hamburgers
At 10 minutes each, the test would last for
$43,200$ minutes $\quad...$ divide with 60 (min. per hour)
= 720 hours $\quad...$ divide with 24 (hours per day)
= 30 days