Quant, Math & Computer Science Puzzles for Interview Preparation & Brain Teasing
A collection of ~225 Puzzles with Solutions (classified by difficulty and topic)

Nov 26, 2012

Multilingual Hedge Fund - Combinatorics Puzzle

Source: Quantnet Forums

Problem:
A hedge fund has 70 employees. For any two employees X and Y there is a language that X speaks but Y does not, and there is a language that Y speaks but X does not. At least how many different languages are spoken by the employees of this hedge fund?

Update: (Dec 17, 2012)
Correct answer posted by Nikhil Jain (IT BHU Alumnus 2008), Paul Wong and Tuhin (IITB 4th year student) in comments.

Similar Problem:
A very similar but more difficult problem posted a couple of years back: http://pratikpoddarcse.blogspot.in/2009/12/number-of-locks-and-keys.html

11 comments:

  1. 8 languages.

    Maximum number of unique combinations possible with 8 languages is 8c4 which is equal to 70. Thus if every employee knows 4 languages then 8 different languages will be sufficient to meet our case.

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  2. 70..each employee speaks diff language...

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  3. let x be number of language, max combination when each one takes up x/2 number of language

    x C (x/2) >= 70

    x = 8

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  4. The least number of languages that can be spoken are 8. Say the set of of languages be L = {A, B, C, .....}. Then for each of 70 employees, there should be a distinct combination for each of the 70 employees. We see that 8 distinct languages suffices because 8!/4!4! = 70, i.e, each of the 70 employees should choose a distinct 4 set from these 8 distinct languages

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  5. 7
    2^7=128 > 70

    Either you know the language (think boolean 1) or you do not (think boolean 0). Number of unique combinations needed is more than 70.

    (baki aap khud samjhdaar hain :P)

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  6. Vaibhav, The question asks for "at least" how many?

    Correct answer is 8. Thanks Nikhil, Paul and Tuhin for the correct answer.


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    Replies
    1. The answer for 71 people should still be 8 but this method gives answer as 9.

      Is this approach right?
      70 people : Take one language. x people speak it and 70-x don't. Hence the total number of languages required = 1 + Max(total_languages(x),total_languages(70-x)). For least number of languages, x = 70 - x. Hence x = 35

      and so on.... after that x= 18.... and then x = 9 ..... then 5... then 3... then 2.... then 1

      This approach gives for 71 people total languages = 8

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    2. This comment has been removed by the author.

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    3. Please Ignore. I messed up the solution.

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  7. This is based on Sperner's Theorem.
    In this case ans is 8.

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