One day Ram and Shyam were searching Google for finding what is the maximum number of hairs on the human head. They found that it is 2, 00, 000. Then Shyam thought a while and suddenly made a statement that there are at least two Bengaluru citizens who have got the exact same number of hairs on their heads. Both of them knew the fact that the population of Bengaluru is 1.23 crores.
But Ram was still thinking whether Shyam’s statement is correct or not.
Is Shyam’s statement 100% correct or we need more information to deduce the truth of the statement?
Shyam’s statement is 100% correct!
Assume you are sorting the citizens according to the number of hairs on their heads. Then, you can imagine virtual bins with these number of hairs on their heads:
0, 1, 2, …., 2, 00, 000 (for a total of 2, 00, 001 bins)
But there are 1.23 crores of them and at least two should be in the same bin.
This problem and solution are based on ‘Pigeonhole principle‘.
There is nothing related to probability in this and Shyam’s statement is 100% correct!
In fact if we assume the total number of people in the city with at most 10, 000 hairs are 23, 00, 000, there will be at least two persons who have same number of hairs, the number being greater than 10, 000 hairs, on their heads. This is because, after taking away 23, 00, 000 people from 1.23 crore, we still have 1 crore people and 1, 90, 000 bins only (namely 10, 001, 10, 002, …, 2, 00, 000)!