Combinatorics and Jason’s Deli (Part 3)

Jason’s Deli is one of my family’s favorite places for an inexpensive meal. Recently, I saw the following placard at our table advertising their salad bar:

The small print says “Math performed by actual rocket scientist”; let’s see how the rocket scientist actually did this calculation.

In yesterday’s post, I showed that the rocket scientist correctly calculated

$\displaystyle {49 \choose 5} = 1,906,884$.

To impress upon customers just how large this number is, the advertisers imagine eating a different salad every day until all 1,906,884 possibilities had been exhausted. Since there are 365 days in a year, apparently the rocket scientist divided:

$\displaystyle \frac{1,906,884}{365} = 5,224.3397...$

Unfortunately, there’s a small problem: the rocket scientist forgot about leap years! Ignoring for now the adjustments of the Gregorian calendar (years divisible by 1000 but not 4000 aren’t leap years — so that 2000 was a leap year but 2100 won’t be), we should divide not by 365 but by 365.25:

$\displaystyle \frac{1,906,884}{365} = 5,220.7638...$

Over a span of 5,220 years, there might be 3 or 4 extra leap days in the above calculation (depending on when someone starts eating the salads), not enough to throw off the above calculation by too much. So the correct answer, rounded to the nearest integer, really should have been 5,221 years.

All this to say, ignoring leap years caused the rocket scientist to give an answer that was off by 3.