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Difference between revisions of "1997 AJHSME Problems/Problem 9"

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Give the students uncreative names:  <math>A</math>, <math>B</math>, and <math>C</math>, replacing the alphabetically first student's name with <math>A</math>, and the alphabetically last student's name with <math>C</math>.  List all the ways they can line up:
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Give the students uncreative names like Abby Adams, Bob Bott, and Carol Crock:  <math>A</math>, <math>B</math>, and <math>C</math>, replacing the alphabetically first student's name with <math>A</math>, and the alphabetically last student's name with <math>C</math>.  List all the ways they can line up:
  
 
<math>ABC</math>
 
<math>ABC</math>
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* [[AJHSME Problems and Solutions]]
 
* [[AJHSME Problems and Solutions]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
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{{MAA Notice}}

Latest revision as of 19:54, 23 October 2015

Problem

Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?

$\text{(A)}\ \dfrac{1}{12} \qquad \text{(B)}\ \dfrac{1}{9} \qquad \text{(C)}\ \dfrac{1}{6} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}$

Solution 1

There are $3$ ways to pick the first student, $2$ ways to pick the second student, and $1$ way to pick the last student, for a total of $3! = 3\times2\times1 = 6$ ways to line the students up.

Only $1$ of those ways is alphabetical. Thus, the probability is $\frac{1}{6}$ or $\boxed{C}$

Solution 2

Give the students uncreative names like Abby Adams, Bob Bott, and Carol Crock: $A$, $B$, and $C$, replacing the alphabetically first student's name with $A$, and the alphabetically last student's name with $C$. List all the ways they can line up:

$ABC$

$ACB$

$BAC$

$BCA$

$CAB$

$CBA$

Only $1$ of those $6$ lists is in alphabetical order, giving an answer of $\frac{1}{6}$ or $\boxed{C}$

See also

1997 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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