## Task B-2.2.

Calculate the arithmetic mean of all four-digit numbers written using the digits $1,3,5,7$, if all digits are used and only once.
## Solution.

There are $4 \cdot 3 \cdot 2 \cdot 1=24$ such four-digit numbers.

1 point

For example, if the digit 1 is in the first position, we have 6 different numbers with the given property: $1357, 1375, 1537, 1573, 1735, 1753$.

Similarly, the remaining numbers can be listed.

It is important to note that each of the four digits will appear exactly 6 times in the thousands, hundreds, tens, and units place.

The sum of all such numbers is:

$$
\begin{gathered}
6(1000+3000+5000+7000)+6(100+300+500+700)+6(10+30+50+70)+6(1+3+5+7) \\
\quad=(6000+600+60+6)(1+3+5+7)=6666 \cdot(1+3+5+7)=6666 \cdot 16 . \quad 2 \text { points }
\end{gathered}
$$

The arithmetic mean of the desired sequence of numbers is $\frac{6666 \cdot 16}{24}=4444 . \quad 1$ point

Note: The desired sum can be calculated as follows:

If we sum all the digits in the units place, we get $6(1+3+5+7)=96$.

The same sum is obtained by summing the tens, hundreds, and thousands places.

Therefore, the total sum of all 24 numbers is $96+960+9600+96000=106656$.