Final answer:
Using the pigeonhole principle, since there are more people in New York City than possible hair counts on a human head, there must be at least nine people with the same number of hairs on their heads.
Step-by-step explanation:
To show that there had to be at least nine people in New York City in 2016 with the same number of hairs on their heads, we can use the pigeonhole principle. This 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. In this context, 'items' are people and 'containers' are the possible number of hairs on their heads.
Let's assume the maximum number of hairs on a human head is 1,000,000. Since there are 8,537,673 people in New York City according to the 2016 population data, and there are only 1,000,000 possible hair counts, applying the pigeonhole principle means that for every distinct number of hairs possible from 0 to 1,000,000, there needs to be at least 8 people with that hair count (since 8,537,673 divided by 1,000,000 gives you a quotient larger than 8).
However, since we can't distribute 8,537,673 evenly into 1,000,000 hair counts, there must be some hair counts with more than 8 people. If we add one more person (making it a total of 9 people) to the hair counts that already have 8 people, for at least one of these hair counts, there will be at least 9 persons. This proves there have to be at least nine people in New York City in 2016 with the same number of hairs on their heads.