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Difference between revisions of "2024 AMC 10B Problems/Problem 12"

m (Protected "2024 AMC 10B Problems/Problem 12" ([Edit=Allow only administrators] (expires 04:59, 14 November 2024 (UTC)) [Move=Allow only administrators] (expires 04:59, 14 November 2024 (UTC))))
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==Solution 1==
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Let's say we have some number of languages. Then each student will speak some amount of those languages, and no two people can have the same combination of languages or else the conditions will no longer be satisfied. Notice that <math>{9}\choose{4}</math> <math>= 126 \geq 100</math>. So each of the <math>100</math> students can speak some <math>4</math> of the <math>9</math> languages. Thus, <math>\boxed{9}</math> is our answer.
  
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~lprado

Revision as of 00:33, 14 November 2024

Solution 1

Let's say we have some number of languages. Then each student will speak some amount of those languages, and no two people can have the same combination of languages or else the conditions will no longer be satisfied. Notice that ${9}\choose{4}$ $= 126 \geq 100$. So each of the $100$ students can speak some $4$ of the $9$ languages. Thus, $\boxed{9}$ is our answer.

~lprado

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