Strings of three decimal digits

Oct 2008
1,116
431
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.
 
Feb 2014
1,748
651
United States
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.
The problem is that you are assuming that you can distinguish between

$4_14_2blank\ and\ 4_24_1blank,$ but they both show up as $44blank.$ You are double counting.

If one 4 was red and the other was blue and you could distinguish between the 4s, your incorrect logic would apply to certain questions.

The simplest way to ponder this question is to ask where the digit different from 4 can go? That is 3 distinguishable spots. And in each spot you have 9 choices so 3 * 9 = 27.
 
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