16 thoughts on “Pigeonhole Principle – 4

  1. This is exactly what I was searching for. I have come across similar questions and was not sure how to solve them. Thanks a lot and keep up the good work 🙂

  2. When cubes there are only 5 possible remainders… How did you find this out ? Is there a method to find this because this could be very helpful during exams. Or is it just trial and error ?

  3. Hi J! I found your post really helpful.. but I am not getting the solution of q1. If it was asked to have the minimum number of boxes with the same number of oranges, then the answer would had been 4 , am I right? Your post probably has an error in the final answer part.. 🙂

    • No, the minimum value would have been 0. There could be 0 boxes having a particular number! But this is a different question – it is asking for a minimax, that is, the minimum value of the maximum possible number. Read the earlier 3 posts again.

      Let me phrase it differently. Suppose I say, you want to find the mode of the distribution, what could be the least possible value for the mode? You will find that it could be 5 but not less.


      • Sir but the minimum possible value for the mode can be 4 right? As 5×31=155. which implies that some weights must have no. of boxes less than 5. Please correct me if I’m wrong.

      • Remember that we are asked the minimum possible value of the maximum. As you said, Some may be less than 5, but do note that all cannot be less than 5. So the maximum is at least 5.


  4. Sir I am not able to distinguish when to use the least integer approach directly and when to think over the worst case scenario. I mean I understand we used the latter in approaches like the “minimum number of items to be picked to guarantee a condition” but when do we use the least integer approach then?

    Also, in the boxes and apples Questions, Can you please tell me why the 126 boxes aren’t our pigeonholes sir? Shouldn’t we distribute 120-150 pigeons into 126 pigeonholes as in the chocolate problem of 75 chocolates among 8 kids?

    • You need to decide on a case-by-case basis. That will come through practice, usually. As for your other question – do you know the total chocolates to be distributed? If those were known (and were your pigeons), the boxes could be your pigeonholes. Here you want to find a limit for the number of boxes having a certain value, not for the number of chocolates in each box.


  5. Hi J,

    Can the following question be solved using the pigeonhole principle??
    A survey about preferred TV channels was conducted among a group of 10,000 people. The following were
    the results:
    93% liked Sony TV,
    89% liked Zee TV,
    81% liked Star Plus,
    75% liked Zee Cinema,
    78% liked MTV,
    and 100 people did not like any of these five channels.
    Find the minimum number of people who like all these five channels.

    • Yes…remove the 100 people ie 1% and the rest have to be 99. Suppose we take 99 x 4 it is 396 but 93 + 89 + 81 + 75 + 78 = 416 so the extra 20% or 2000 should probably be the answer. (Edited for misreading)


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