Derek's phone number, $336$ - $7624,$ has the property that the three-digit prefix, $336,$ equals the product of the last four digits, $7 / times 6 / times 2 / times 4.$ how many seven-digit phone numbers beginning with $336$ have this property?

There are 96 different numbers with this property.

The prime factorization of 336 is 2⁴*3*7. This means the different numbers used could be:

2, 4, 6, 7 (as in the one listed);

2, 3, 8, 7;

3, 4, 4, 7;

1, 6, 7, 8.

For each one of these, there are 4! permutations (where order is important). 4!=24.

4 different sets of numbers multiplied by 24 permutations = 96 choices.