To enter her secret lab, SuperMathHeroine must punch in a 7-digit code, where each digit can be from 0 to 9. Unfortunately, SuperMathHeroine has forgotten her code. She only remembers that the product of the digits in her code is 10,000. How many different codes could there be?
See the answer here:
Well, none of the digits can be 0 because the product is than going to be 0. You can't have more than 2 1's either.
To get started, we turn to the factorization of 10,000. It factors out as 2^4 * 5^4. This means that the only possible combinations of numbers are:
1,2,8,5,5,5,5 -> 105 ways to arrange this
2,2,4,5,5,5,5 -> 105 ways to arrange this
1,4,4,5,5,5,5 -> 105 ways to arrange this
1,1,8,5,5,5,5 -> 105 wasy to arrange this
So the number of different possible codes is 105+105+105+105 = 420 possible codes.