Difference between revisions of "1990 AIME Problems/Problem 8"
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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> | 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> | ||
− | Since the | + | 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> |
~Azjps (Solution) | ~Azjps (Solution) | ||
Line 26: | Line 26: | ||
<li>There are <math>8!</math> ways to arrange <math>8</math> distinguishable letters. However: <p> | <li>There are <math>8!</math> ways to arrange <math>8</math> distinguishable letters. However: <p> | ||
<ul style="list-style-type:square;"> | <ul style="list-style-type:square;"> | ||
− | <li>Since there are <math>3</math> indistinguishable <math>L</math>'s, we have counted each distinct arrangement of the <math>L</math>'s <math>3!</math> times. </li><p> | + | <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> |
− | <li>Since there are <math>2</math> indistinguishable <math>M</math>'s, we have counted each distinct arrangement of the <math>M</math>'s <math>2!</math> times. </li><p> | + | <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> |
− | <li>Since there are <math>3</math> indistinguishable <math>R</math>'s, we have counted each distinct arrangement of the <math>R</math>'s <math>3!</math> times. </li><p> | + | <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> |
</ul> | </ul> | ||
− | By the Multiplication Principle, we have counted each distinct arrangement of <math>LLLMMRRR</math> <math>3!\cdot2!\cdot3!</math> times. <p> | + | By the Multiplication Principle, we have counted each distinct arrangement of <math>LLLMMRRR</math> for <math>3!\cdot2!\cdot3!</math> times. <p> |
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> | 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> | ||
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> | 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> | ||
− | <li>First, we have <math>\binom83</math> ways to choose any <math>3</math> from the <math>8</math> | + | <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> |
− | Next, we have <math>\binom52</math> ways to choose any <math>2</math> from the <math>5</math> remaining | + | 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> |
− | Finally, we have <math>\binom33</math> way to choose <math>3</math> from the <math>3</math> remaining | + | 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> |
By the Multiplication Principle, the answer is <math>\binom83\binom52\binom33=560.</math> | By the Multiplication Principle, the answer is <math>\binom83\binom52\binom33=560.</math> | ||
</li><p> | </li><p> |
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?
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 and respectively. We consider the string
Since the letter arrangements of and the shooting orders have one-to-one correspondence, we count the letter arrangements:
~Azjps (Solution)
~MRENTHUSIASM (Revision)
Remark
We can count the letter arrangements of in two ways:
- There are ways to arrange distinguishable letters. However:
- Since there are indistinguishable 's, we have counted each distinct arrangement of the 's for times.
- Since there are indistinguishable 's, we have counted each distinct arrangement of the 's for times.
- Since there are indistinguishable 's, we have counted each distinct arrangement of the 's for times.
As shown in the Solution section, we use division to fix the overcount. The answer is
Alternatively, we can use a multinomial coefficient to obtain the answer directly:
- First, we have ways to choose any from the positions for the 's.
Next, we have ways to choose any from the remaining positions for the 's.
Finally, we have way to choose from the remaining positions for the 's.
By the Multiplication Principle, the answer is
~MRENTHUSIASM
Video Solution
https://www.youtube.com/watch?v=NGfMLCRUs3c&t=7s ~ MathEx
See also
1990 AIME (Problems • Answer Key • Resources) | ||
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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