Question

9. Show that In a packed Carnegie Hall performance, there will be two people who have the same first and last initials. (The main hall at Carnegie seats roughly 2,800)

10. Imagine a certain college has 6,000 American students, at least one from each of the 50 states. Show there must be a group of 120 students coming from same state.

Answer 1

According to More generalised version of Pigeon Hole principle "The maximum value is at least the average value, for any non-empty finite bag of real numbers"

9) Total No. of People are 2800.

In english we have 26 alphabet meansing there will be 26x26 = 676 possiblities for first and last initials. Now consider 676 possibilities as 676 pigeon holes. Considers 2800 people as pigeons. Now we need to fit 2800 pigeons into 676 holes. We cannot definitely place each pigeon in each pigeon hole because total pigeons are more than 676. So according to pigeon hole principle we can find a atleast one  hole which contains two pigeons with same first and last initials

10) Given that 6000 American students come from 50 states. So average would be 6000/50 = 120. So according to generalised pigeon hole principles there must atleast the average value which is 120 students coming from the same state.