Difference between revisions of "1990 AIME Problems/Problem 8"

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== Problem ==
 
== Problem ==
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In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:
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1) The marksman first chooses a column from which a target is to be broken.
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2) The marksman must then break the lowest remaining target in the chosen column.
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If the rules are followed, in how many different orders can the eight targets be broken?
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[[File:1990 AIME Problem 8.png|center|160px]]
  
 
== Solution ==
 
== Solution ==
{{solution}}
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Clearly, the marksman must shoot the left column three times, the middle column two times, and the right column three times.
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From left to right, suppose that the columns are labeled <math>L,M,</math> and <math>R,</math> respectively. We consider the string <math>LLLMMRRR:</math>
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Since the letter arrangements of <math>LLLMMRRR</math> and the shooting orders have one-to-one correspondence, we count the letter arrangements: <cmath>\frac{8!}{3!\cdot2!\cdot3!} = \boxed{560}.</cmath>
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~Azjps (Solution)
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~MRENTHUSIASM (Revision)
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== Remark ==
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We can count the letter arrangements of <math>LLLMMRRR</math> in two ways:
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<ol style="margin-left: 1.5em;">
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  <li>There are <math>8!</math> ways to arrange <math>8</math> distinguishable letters. However: <p>
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<ul style="list-style-type:square;">
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<li>Since there are <math>3</math> indistinguishable <math>L</math>'s, we have counted each distinct arrangement of the <math>L</math>'s for <math>3!</math> times. </li><p>
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<li>Since there are <math>2</math> indistinguishable <math>M</math>'s, we have counted each distinct arrangement of the <math>M</math>'s for <math>2!</math> times. </li><p>
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<li>Since there are <math>3</math> indistinguishable <math>R</math>'s, we have counted each distinct arrangement of the <math>R</math>'s for <math>3!</math> times. </li><p>
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</ul>
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By the Multiplication Principle, we have counted each distinct arrangement of <math>LLLMMRRR</math> for <math>3!\cdot2!\cdot3!</math> times. <p>
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As shown in the <b>Solution</b> section, we use division to fix the overcount. The answer is <math>\frac{8!}{3!\cdot2!\cdot3!} = 560.</math> <p>
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Alternatively, we can use a multinomial coefficient to obtain the answer directly: <math>\binom{8}{3,2,3}=\frac{8!}{3!\cdot2!\cdot3!} = 560.</math> </li><p>
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  <li>First, we have <math>\binom83</math> ways to choose any <math>3</math> from the <math>8</math> positions for the <math>L</math>'s.<p>
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Next, we have <math>\binom52</math> ways to choose any <math>2</math> from the <math>5</math> remaining positions for the <math>M</math>'s.<p>
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Finally, we have <math>\binom33</math> way to choose <math>3</math> from the <math>3</math> remaining positions for the <math>R</math>'s.<p>
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By the Multiplication Principle, the answer is <math>\binom83\binom52\binom33=560.</math>
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</li><p>
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</ol>
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~MRENTHUSIASM
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==Video Solution==
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https://www.youtube.com/watch?v=NGfMLCRUs3c&t=7s ~ MathEx
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== See also ==
 
== See also ==
* [[1990 AIME Problems]]
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{{AIME box|year=1990|num-b=7|num-a=9}}
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[[Category:Intermediate Combinatorics Problems]]
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{{MAA Notice}}

Latest revision as of 17:44, 4 September 2021

Problem

In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:

1) The marksman first chooses a column from which a target is to be broken.

2) The marksman must then break the lowest remaining target in the chosen column.

If the rules are followed, in how many different orders can the eight targets be broken?

1990 AIME Problem 8.png

Solution

Clearly, the marksman must shoot the left column three times, the middle column two times, and the right column three times.

From left to right, suppose that the columns are labeled $L,M,$ and $R,$ respectively. We consider the string $LLLMMRRR:$

Since the letter arrangements of $LLLMMRRR$ and the shooting orders have one-to-one correspondence, we count the letter arrangements: \[\frac{8!}{3!\cdot2!\cdot3!} = \boxed{560}.\]

~Azjps (Solution)

~MRENTHUSIASM (Revision)

Remark

We can count the letter arrangements of $LLLMMRRR$ in two ways:

  1. There are $8!$ ways to arrange $8$ distinguishable letters. However:

    • Since there are $3$ indistinguishable $L$'s, we have counted each distinct arrangement of the $L$'s for $3!$ times.
    • Since there are $2$ indistinguishable $M$'s, we have counted each distinct arrangement of the $M$'s for $2!$ times.
    • Since there are $3$ indistinguishable $R$'s, we have counted each distinct arrangement of the $R$'s for $3!$ times.
    By the Multiplication Principle, we have counted each distinct arrangement of $LLLMMRRR$ for $3!\cdot2!\cdot3!$ times.

    As shown in the Solution section, we use division to fix the overcount. The answer is $\frac{8!}{3!\cdot2!\cdot3!} = 560.$

    Alternatively, we can use a multinomial coefficient to obtain the answer directly: $\binom{8}{3,2,3}=\frac{8!}{3!\cdot2!\cdot3!} = 560.$

  2. First, we have $\binom83$ ways to choose any $3$ from the $8$ positions for the $L$'s.

    Next, we have $\binom52$ ways to choose any $2$ from the $5$ remaining positions for the $M$'s.

    Finally, we have $\binom33$ way to choose $3$ from the $3$ remaining positions for the $R$'s.

    By the Multiplication Principle, the answer is $\binom83\binom52\binom33=560.$

~MRENTHUSIASM

Video Solution

https://www.youtube.com/watch?v=NGfMLCRUs3c&t=7s ~ MathEx

See also

1990 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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