+0  
 
+9
104
3
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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?

 Jan 20, 2020
 #1
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0

See the answer here:

 

https://web2.0calc.com/questions/how-many-ways-are-there-to-form-a-7-digit-code-where

 Jan 20, 2020
 #2
avatar+313 
+1

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.

 Jan 20, 2020
edited by ThatOnePerson  Jan 20, 2020
 #3
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Some of your permutations are wrong:

 

1,2,8,5,5,5,5 -> 105 ways to arrange this >>This one is 7!/4! =210 ways

 

1,1,8,5,5,5,5 -> 105 wasy to arrange this>> Here the permutations are correct but the product is =5,000.

 Jan 20, 2020

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