Difference between revisions of "2024 AMC 10B Problems/Problem 17"
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+ | ==Solution 1== | ||
+ | We perform casework based on how many people tie. Let's say we're dealing with the following people: <math>A,B,C,D,E</math>. | ||
+ | <math>5</math> people tied: All <math>5</math> people tied for <math>1</math>st place, so only <math>1</math> way. | ||
+ | |||
+ | <math>4</math> people tied: <math>A,B,C,D</math> all tied, and <math>E</math> either got <math>1</math>st or last. <math>{5}\choose{1}</math> ways to choose who isn't involved in the tie and <math>2</math> ways to choose if that person gets first or last, so <math>10</math> ways. | ||
+ | |||
+ | <math>3</math> people 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 people not involved in the tie. So <math>6 \cdot 10 = 60</math> ways. | ||
+ | |||
+ | <math>2</math> people 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 people not involved in the tie. So <math>24 \cdot 10 = 240</math> ways. | ||
+ | |||
+ | It's impossible to have "1 person tie", so that case has <math>0</math> ways. | ||
+ | |||
+ | Finally, there are no ties. We just arrange the <math>5</math> people, so <math>5! = 120</math> ways. | ||
+ | |||
+ | The answer is <math>1+10+60+240+0+120 = \boxed{431}</math>. | ||
+ | |||
+ | ~lprado |
Revision as of 00:15, 14 November 2024
Solution 1
We perform casework based on how many people tie. Let's say we're dealing with the following people: .
people tied: All people tied for st place, so only way.
people 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 person gets first or last, so ways.
people tied: We have . There are ways to determine the ranking of the groups. There are ways to determine the two people not involved in the tie. So ways.
people tied: We have . There are ways to determine the ranking of the groups. There are ways to determine the three people not involved in the tie. So ways.
It's impossible to have "1 person tie", so that case has ways.
Finally, there are no ties. We just arrange the people, so ways.
The answer is .
~lprado