Answer:
There are 64,000 different combinations
for the lock. 

Complete Solution:
In the challenge, the lock uses
the numbers 0 to 39. Start with an easier problem using
1, 2, and 3. You can count the possibilities by drawing
a tree diagram. A portion of this tree diagram is shown
below:
If you start with
1, you get nine different combinations. If you start with
2 instead of 1, you also get nine different combinations.
If you start with 3, you get nine more possibilities for
a total of 9 + 9 + 9 = 27 different
combinations.
Think about this
in terms of choices.
You can choose any of three numbers as a possible first
number in the combination, follow that with a choice of
any of the three numbers as the second number in the combination,
and finally choose any of the three numbers for the final
number in the combination for a total of 3 ×
3 ×
3, or 27 different combinations.
If the lock used the numbers 1, 2, 3, and
4, you would have to choose from four
numbers, three different times. This would
give you 4 ×
4 ×
4, or 64 choices for the
combination. The lock in the challenge requires that you
choose from 40 different numbers,
three different times. Therefore, there are
40 ×
40 ×
40, or 64,000 different combinations. 