2005 AMC 10B Problem 20

Question

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?

48000

49999.5

53332.8

55555

56432.8

By symmetry, each digit appears equally often in each place

The average digit in each place is

\dfrac{1+3+5+7+8}{5},

then multiply by

11111

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.

Problem 20 in other years