2005 AMC 10B Problem 20

What is the average (mean) of all

5

-digit numbers that can be formed by using each of the digits

1,

3,

5,

7,

and

8

exactly once?

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Solution

By symmetry, each of the five digits appears equally often in each place, so the average digit in every place is

\dfrac{1 + 3 + 5 + 7 + 8}{5} = 4.8.

The average number is therefore

\begin{gathered} \small 4.8(1 + 10 + 100 + 1000 + 10000) \\ = 4.8 \cdot 11111 = 53332.8. \end{gathered}

Thus, C is the correct answer.

Tagged: permutations · place value · symmetry