[ Counting in two ways $]$ Counting in two ways $]$

Vasya thought of three different non-zero digits. Petya wrote down all possible two-digit numbers, in the decimal representation of which only these digits were used. The sum of the written numbers is 231. Find the digits thought of by Vasya.
Let a, $b$ and c be three digits thought of by Vasya. There are nine two-digit numbers, in the decimal representation of which only these digits are used: ; ; ; ; ; ; ; . Let's find their sum, by decomposing each number into a sum of place value addends: $(10 a+a)+(10 b+b)+(10 c+c)+(10 a+b)+(10 b+a)+$ $(10 a+c)+(10 c+a)+(10 b+c)+(10 c+b)=33 a+33 b+33 c=33(a+b+c)$. According to the condition, $33(a+b+c)=231$, that is, $a+b+c=7$. There is a unique triplet of different and non-zero digits, the sum of which is 7.

## Answer

Vasya thought of the digits $1 ; 2$ and 4.