MadeEasy Test Series: Combinatory - Pigeonhole Principle

Question

A community of 5 members is to be formed out of 10 people. The names are written in chits of paper and put into 6 boxes. So how many chits will go into the same box?

 

Anyone, please make me understand this question.

Comments

is it 43?

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no of chits  10c5 =252

pigeonhole principle t floor[252-1/6] = 42

Answer 1

This question basically uses Pigeon Hole Principle and combination(selection) formula.According to pigeon hole principle :

If  we have m objects and n holes , then we can say at least ceil(m/n) objects will be allocated the same hole.

So the keywords to look into for such problems are "at least how many" and "same location or same box etc"

So now to calculate number of objects which is number of chits possible here each of which comprises of 5 names to be selected out of 10 persons.

So no of chits   =   No of ways of selecting 5 persons from 10 persons

                       =   ¹⁰C₅

                       =   (10 * 9 * 8 * 7 * 6) / 120

                       =   252

So  we have value of m in this context = 252

And no of holes given which is no of boxes in this case  =  6

So no of chits which will go to same box   =  ceil(252 / 6)

                                                             =   42

Hence at least 42 chits will go into same box.

Comments

@Habib. I think the question is asking something different.

Its saying that the names of 10 persons are written on 10 chits and chits are put into 6 boxes. So, there will be atleast 2 chits that will go to the same box.

So, I think answer should be 2.

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@sushant  habib answer is correct and i agree...

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No there will not be 10 chits..As an individual chit contains the names of 5 people out of 10 people..So no of chits is nothing but no of ways we can select 5 people out of 10 people , as for one selection we will have 1 corresponding chit..

Thus there will be ¹⁰C₅ = 252 chits..Hence ceil(252/6) = 42 is the correct answer..

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But why 252 chits? where does 252 chits come from? They have just said names are written on chits of paper.

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@Habib. Thats your interpretation. But its difficult to interpret from statement " .... The names are written in chits of paper and put into 6 boxes ." what exactly is happening

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Rather it depends upon how many members are there in selection committee.

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U have to think about names of whom is written in chit..It is made clear from the 1st statement of the question..

And why are u thinking about selection committee ??

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Because      no of chits  = no  of committee members

I slightly disagree with your reasoning for your interpretation.

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It is not like that bro..Chits contain the name of 5 people out of 10 people..And hence likewise we will have ¹⁰C₅ chits..

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But why 6 boxes?

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The boxes will be there to place the chits in..:)..So ¹⁰C₅ chits will get placed in one of the boxes..So the chits are acting like pigeons and boxes are acting like holes..

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Yes, but there should be a meaning for 6 rt?

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@Arjun, It seems to me that Pigeonhole has no meaning for the above question. Still I am not getting  why Habib interpreted it like this i.e it is not the same interpretation if 100 GATE aspirants view this question. If its same interpretation for atleast more than 50%, I can understand that I am going wrong.

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i also got answer as 2:(

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lol my answer was also 2

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as simple as that we have to find n and m , rest the question is in our hand.. so to find m first, m is given.. always remember m is the value that is the pigeonhole are the value that is given after the SAME keyword and the value of n we need to find from the combinaton of 10 and 5, that is 252

at last we divide [n/m] and we have our answer that is 42!