A three digit entry code

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Question from Bryon, a parent:

The entry code for the executive bathroom is a three-digit odd number
with no repeated digits. If you want to try to break in, what is the
maximum number of numbers you'll have to try?
(Remember that the three-digit numbers cannot start with zero.)

Byron,

Since the number is odd there are 5 choices for the units digit, 1, 3, 5, 7 and 9. If you put a zero in the tens digit then there are 8 choices for the hundreds digit (not 0 and not the units digit) giving 8 x 1 x 5 possible combinations with a zero in the tens digit. If the tens digit is not zero then there are 8 choices for the tens digit and now only 7 choices for the hundreds digit giving 7 x 8 x 5 more possibilities. Thus the number of possible combinations is 8 x 1 x 5 + 7 x 8 x 5.

Penny and Steve

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