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Adding Digits

Problem

Imagine writing down every possible combination of the digits 1, 2, 3, and 4. What would be the sum of all these combinations?

For example, with the digits 1, 2 and 3; there are six combinations and the sum of these combinations is
123 + 132 + 213 + 231 + 312 + 321 = 1332.


Solution

Considering combinations,

1234Each digit appears in each column 6 times.
1243So each column must add up to,
13246(1 + 2 + 3 + 4) = 60.
1342 
1423therefore Total = 60(1 + 10 + 100 + 100) = 66660.
1432 
2134 , etc. 

What about the digits 1, 2, 3, ..., n ?

Problem ID: 109 (Mar 2003)     Difficulty: 2 Star

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