MHB Strings of three decimal digits

[alexmahone]

How many strings of three decimal digits have exactly two digits that are 4s?

Solution:

44x - 9 strings
4x4 - 9 strings
x44 - 9 strings
Total: 27 strings

I get that.

But what is wrong with this line of reasoning? I have two 4s with me. I have 3 positions for the first one and 2 for the second one. Then I will have a blank space that can be filled by one of 9 digits. So, the total number of strings is 3*2*9=54.

 

[Nono713]

Because with that reasoning you can place your first 4 in position 1 and your second 4 in position 2 and let the remaining digit be, say, 5, giving 445, but you can also place your first 4 in position 2, your second 4 in position 1, giving you... 445. So your approach counts the same solution twice, because it's taking into account order of the 4's when that order doesn't matter. Hence the number of solutions is actually 54/2 = 27.

 

[mrtwhs]

How many strings of three decimal digits have exactly two digits that are 4s?

Solution:

44x - 9 strings
4x4 - 9 strings
x44 - 9 strings
Total: 27 strings

I get that.

But what is wrong with this line of reasoning? I have two 4s with me. I have 3 positions for the first one and 2 for the second one. Then I will have a blank space that can be filled by one of 9 digits. So, the total number of strings is 3*2*9=54.

But you can't distinguish the 4's. There is no way to tell the "first four" from the "second four". You would be correct if one of the fours was red and the other four was green.

44x
44x
4x4
4x4
x44
x44

But if they are not colored then you only have three ways.