37 diplomats from New Zealand, Australia, Fiji, Tonga, Samoa and the Cook Islands went out to dinner together after a South Pacific Forum meeting. Can we be certain that one country was represented by at least 7 diplomats? (i) First identify the pigeonholes: The countries (What they had for dinner The diplomats (ii) The answer: Yes we can be certain O No we can\'t be certain Consider the pigeonholes {1, 99}, {3,97}, {5,95},... {49, 51} - i.e. all pairs of positive odd numbers adding to 100. Can we be certain that in any set of 25 different positive odd numbers, all less than 100, there is a pair of numbers whose sum is 100. Consider the pigeonholes {1}, {3,99}, {5,97},... {49,53}, {51}-which contain all positive odd numbers less than 100. Two of these are \"singletons\" containing just one number, while the remainder contain a pair adding to 102. Can we be certain that in any set of 27 different positive odd numbers, all less than 100, there is a pair of numbers whose sum is 102? Solution The pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item (a) (i) Here Diplomats are the pigeons (ii)No,we cannot be certain that one country would be represented by atleat 7 diplomats but we can be certain that atleast one country would be having more than one diplomat. (b) Yes,we can be sure of that. (c) No, we cannot be absolutely sure of that because we can have 27 different positive odd numbers which does not include any pair whose sum is 102..

1. 37 diplomats from New Zealand, Australia, Fiji, Tonga, Samoa and the Cook Islands went out to
dinner together after a South Pacific Forum meeting. Can we be certain that one country was
represented by at least 7 diplomats? (i) First identify the pigeonholes: The countries (What they
had for dinner The diplomats (ii) The answer: Yes we can be certain O No we can't be certain
Consider the pigeonholes {1, 99}, {3,97}, {5,95},... {49, 51} - i.e. all pairs of positive odd
numbers adding to 100. Can we be certain that in any set of 25 different positive odd numbers,
all less than 100, there is a pair of numbers whose sum is 100. Consider the pigeonholes {1},
{3,99}, {5,97},... {49,53}, {51}-which contain all positive odd numbers less than 100. Two of
these are "singletons" containing just one number, while the remainder contain a pair adding to
102. Can we be certain that in any set of 27 different positive odd numbers, all less than 100,
there is a pair of numbers whose sum is 102?
Solution
The pigeonhole principle states that if n items are put into m containers, with n > m, then at least
one container must contain more than one item
(a)
(i) Here Diplomats are the pigeons
(ii)No,we cannot be certain that one country would be represented by atleat 7 diplomats but we
can be certain that atleast one country would be having more than one diplomat.
(b) Yes,we can be sure of that.
(c) No, we cannot be absolutely sure of that because we can have 27 different positive odd
numbers which does not include any pair whose sum is 102.