2002 AMC 10B Problem 9

Question

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

Words starting with

A,

M,

O,

or

S

come first; count them

Among the words starting with

U,

find how many precede

USAMO

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.

Problem 9 in other years