Question 1199154: Consider all of the ways of ordering the 5 letters U,S,A,M,O. Suppose these are listed in a dictionary in alphabetical order, starting with AMOSU. What position In the list would USAMO be?
A. 97 B. 103 C. 109 D. 115 E. 120
(If your answer is not one of these answer choices, you should double check your work, as it is wrong.)
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website! .
Consider all of the ways of ordering the 5 letters U,S,A,M,O.
Suppose these are listed in a dictionary in alphabetical order, starting with AMOSU.
What position In the list would USAMO be?
A. 97 B. 103 C. 109 D. 115 E. 120
(If your answer is not one of these answer choices, you should double check your work, as it is wrong.)
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Consider the list of all alphabetically ordered words of the 5 letters U,S,A,M,O.
Before than U will appear in the 1st positions, all the words with U in the
5th position, 4th position, 3rd position and 2nd position must appear.
The number of words with U in the 5th position is 4! = 24.
The same is the number of words with U in the 4th position;
3rd position and
2nd positions.
It gives 4*4! = 4*24 = 96 such preceding words.
Next, with U in the 1st position, before than S will appear in the 2nd position,
all the words, starting with U and with S in the 5th position, 4th position and
3rd position must appear. The number of such words is 3*3! = 3*6 = 18.
As soon as we have the word starting with "US" in the list, the next word will be USAMO.
Its sequential number in the list is 96 + 18 + 1 = 115.
ANSWER. The position of the word USAMO in the list is 115.
Solved, with full explanations.
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USAMO is the abbteviation for "The United States of America Mathematical Olympiad".
The United States of America Mathematical Olympiad (USAMO) is a highly selective
high school mathematics competition held annually in the United States.
Since its debut in 1972, it has served as the final round of the American Mathematics Competitions.
See this Wikipedia article
https://en.wikipedia.org/wiki/United_States_of_America_Mathematical_Olympiad#:~:text=The%20United%20States%20of%20America,of%20the%20American%20Mathematics%20Competitions.
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
The number of arrangements of the 5 letters is 5! = 120. We are looking for the position in an alphabetical list of the 120 arrangements of "USAMO".
In the list of 120 arrangements, there are 4! = 24 arrangements starting with each letter. Since U is the last of the 5 letters alphabetically, there are 4*24 = 96 arrangements in the list before the first one with first letter U.
Given first letter U, there are 3! = 6 arrangements with each of the 4 second letters. Among the letters S, A, M, and O, S is last alphabetically, so there are 3*6=18 arrangements with first letter U before the first one with second letter S.
So, before the first arrangement with first two letters US, there are 96 + 18 = 114 arrangements.
Among the 6 arrangements with first two letters US, the arrangement USAMO is first alphabetically, so the position in the list of USAMO is 114 + 1 = 115.
ANSWER: 115
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