2002 AMC 10B Problem 9

Below is the professionally curated solution for Problem 9 of the 2002 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 10B solutions, or check the answer key.

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Concepts:permutations systematic listing

Difficulty rating: 1280

Using the letters $A,$ $M,$ $O,$ $S,$ and $U,$ we can form $120$ five-letter “words.” If these “words” are arranged in alphabetical order, then the “word” $USAMO$ occupies which position?

$112$

$113$

$114$

$115$

$116$

Solution:

The alphabetical order of the letters is $A, M, O, S, U.$ Words beginning with $A,$ $M,$ $O,$ or $S$ fill positions $1$ through $96$ (four choices of first letter, $24$ each).

Words beginning with $U$ occupy positions $97$ – $120.$ Listing them alphabetically, $USAMO$ is the $19$ th such word, so it occupies position $96 + 19 = 115.$

Thus, the correct answer is D.

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