Difference between revisions of "2024 AMC 10B Problems/Problem 17"
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<math>3</math> snails tied: We have <math>ABC, D, E</math>. There are <math>3! = 6</math> ways to determine the ranking of the <math>3</math> groups. There are <math>5\choose2</math> ways to determine the two snails not involved in the tie. So <math>6 \cdot 10 = 60</math> ways. | <math>3</math> snails tied: We have <math>ABC, D, E</math>. There are <math>3! = 6</math> ways to determine the ranking of the <math>3</math> groups. There are <math>5\choose2</math> ways to determine the two snails not involved in the tie. So <math>6 \cdot 10 = 60</math> ways. | ||
− | <math>2</math> snails tied: We have <math>AB, C, D, E</math>. There are <math>4! = 24</math> ways to determine the ranking of the <math>4</math> groups. There are <math>5\choose{3}</math> ways to determine the three | + | <math>2</math> snails tied: We have <math>AB, C, D, E</math>. There are <math>4! = 24</math> ways to determine the ranking of the <math>4</math> groups. There are <math>5\choose{3}</math> ways to determine the three snails not involved in the tie. So <math>24 \cdot 10 = 240</math> ways. |
It's impossible to have "1 snail tie", so that case has <math>0</math> ways. | It's impossible to have "1 snail tie", so that case has <math>0</math> ways. | ||
− | Finally, there are no ties. We just arrange the <math>5</math> | + | Finally, there are no ties. We just arrange the <math>5</math> snails, so <math>5! = 120</math> ways. |
− | The answer is <math>1+10+60+240+0+120 = \boxed{431}</math>. | + | The answer is <math>1+10+60+240+0+120 = \boxed{\text{(D) }431}</math>. |
~lprado | ~lprado | ||
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The answer is <math>120+240+60+10+1=\boxed{\text{(D) }431}</math> ~Tacos_are_yummy_1 | The answer is <math>120+240+60+10+1=\boxed{\text{(D) }431}</math> ~Tacos_are_yummy_1 | ||
+ | |||
+ | ==Solution 3 (Get Lucky)== | ||
+ | For the case of a 5-way tie, we have <math>\dbinom{5}{5}=1</math> cases. We can assume this leads to an answer that ends in 1, leaving only <math>B</math> and <math>D</math>. By just accounting for the cases for a 1-way tie (<math>5!=120</math>) and 2-way tie (<math>\dbinom{5}{2}\cdot4\cdot3!=240</math>), it immediately shows that <math>B</math> does not work. Therefore, the answer is <math>\boxed{\text{(D) }431}</math> | ||
+ | ~shreyan.chethan | ||
+ | |||
+ | Notice that with 2 remaining choices, through guessing, the expected value of your points is <math>\frac{6}{2}=3</math>, and not answering gives 1.5 points. Therefore, you gain an expected value of <math>1.5</math> points by answering. It would be best to always guess whenever you can get rid of <math>2</math> or more answers. If you can only eliminate <math>1</math> answer, not guessing will give you <math>1.5</math> points, but guessing also gives you an expected value of <math>1.5</math> points. I usually leave it blank if I can only get rid of <math>1</math> answer choice. | ||
+ | ~shreyan.chethan, edited by BenjaminDong01, last part added by PerseverePlayer | ||
==Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)== | ==Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)== | ||
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==Video Solution 2 by SpreadTheMathLove== | ==Video Solution 2 by SpreadTheMathLove== | ||
− | https://www.youtube.com/watch?v= | + | https://www.youtube.com/watch?v=Q7fwWZ89MC8 |
==See also== | ==See also== | ||
{{AMC10 box|year=2024|ab=B|num-b=16|num-a=18}} | {{AMC10 box|year=2024|ab=B|num-b=16|num-a=18}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 21:39, 23 November 2024
Contents
Problem
In a race among snails, there is at most one tie, but that tie can involve any number of snails. For example, the result might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?
Solution 1
We perform casework based on how many snails tie. Let's say we're dealing with the following snails: .
snails tied: All snails tied for st place, so only way.
snails tied: all tied, and either got st or last. ways to choose who isn't involved in the tie and ways to choose if that snail gets first or last, so ways.
snails tied: We have . There are ways to determine the ranking of the groups. There are ways to determine the two snails not involved in the tie. So ways.
snails tied: We have . There are ways to determine the ranking of the groups. There are ways to determine the three snails not involved in the tie. So ways.
It's impossible to have "1 snail tie", so that case has ways.
Finally, there are no ties. We just arrange the snails, so ways.
The answer is .
~lprado
Solution 2 (Solution 1 but less words)
Split the problem into cases. A tie of snails has ways to choose the snails that are tied, ways to choose which place they tie for, and to place the remaining snails.
1. No tie
2. Tie of 2 snails
3. Tie of 3 snails
4. Tie of 4 snails
5. Tie of all 5 snails
The answer is ~Tacos_are_yummy_1
Solution 3 (Get Lucky)
For the case of a 5-way tie, we have cases. We can assume this leads to an answer that ends in 1, leaving only and . By just accounting for the cases for a 1-way tie () and 2-way tie (), it immediately shows that does not work. Therefore, the answer is ~shreyan.chethan
Notice that with 2 remaining choices, through guessing, the expected value of your points is , and not answering gives 1.5 points. Therefore, you gain an expected value of points by answering. It would be best to always guess whenever you can get rid of or more answers. If you can only eliminate answer, not guessing will give you points, but guessing also gives you an expected value of points. I usually leave it blank if I can only get rid of answer choice. ~shreyan.chethan, edited by BenjaminDong01, last part added by PerseverePlayer
Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)
https://youtu.be/c6nhclB5V1w?feature=shared
~ Pi Academy
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=Q7fwWZ89MC8
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.