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There are $60$ different 3-letter combinations of A,B,C,D,E. Using each letter to represent a different integer from $1$ to $9$, what is the solution for ABCDE to equal the sum of all 3-letter combinations (ABC + ABD + ABE + BAC + BAD + . . .)?

As well as the correct solution, the best answer should include simplified expression for this problem.

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  • $\begingroup$ To clarify: ABC means the concatenation of A, B and C, not the product of A, B and C, correct? $\endgroup$ – No. 7892142 Feb 13 '15 at 9:31
  • $\begingroup$ @No.7892142 - Correct (a cryptarithm). $\endgroup$ – Len Feb 13 '15 at 10:45
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First let's look at what all the 60 combinations look like. Consider the first digit - obviously, we have 5 choices for which letter to place there. Since there are a total of 60 combinations, each letter must appear there a total of 12 times. Therefore, we can add up everything contributing to the sum from just that place alone as 1200*(A+B+C+D+E). This argument can be applied to each digit separately, so we can evaluate the entire sum as 1332*(A+B+C+D+E) since 12+120+1200=1332.

Now, the constraint that each letter must represent a distinct number means that the sum A+B+C+D+E must fall somewhere in the range of 15 (1+2+3+4+5) to 35 (9+8+7+6+5), inclusive. That's actually a small enough range to check by hand, in which case you get

35964

as the unique answer for this problem.

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When you get $ABCDE=1332(A+B+C+D+E)$, there is a shorter way than checking all possible sums for $A+B+C+D+E$, which, as @SpectralFlame pointed out, can go from 15 to 35. Note that:

$9 \mid 1332$, so $9 \mid ABCDE$. Because a number is divisible by 9 iff the sum of the digits is divisible by $9$, this gives $9 \mid A+B+C+D+E$.

Therefore:

We only have to check $A+B+C+D+E=18$ and $A+B+C+D+E=27$, which give $23976$ (whose digit sum is 27) and 35964 (whose digit sum is indeed 27), so the latter is indeed the only solution.

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