2005 AMC 10B Problem 20

Attempt Problem 20 of the 2005 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 10B solutions, or check the answer key.

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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?

$48000$

$49999.5$

$53332.8$

$55555$

$56432.8$

Answer: C

Concepts:permutations place value symmetry

Difficulty rating: 1540

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.

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