5 Fridays, 5 Saturdays, and 5 Sundays in a month – happens only every 823 years?

If you own an email, Twitter, Facebook, Reddit or now Steemit account, chances are that you have come across an email or a post that claims that an upcoming month has a very rare specific combination of days. A recent version goes something like this:
January 2016 will have 5 Fridays, 5 Saturdays, and 5 Sundays. This happens only once every 823 years. The Chinese call it silver pockets full.
The email is right in that January 2016 had 5 Fridays, 5 Saturdays, and 5 Sundays. But this is nothing special or unusual. You will not have to wait another 823 years for the next time this happens in the month of January. Only 6 years later, in 2021, January will also have 5 Fridays, 5 Saturdays, and 5 Sundays.
In fact, a calendar month that contains 5 Fridays, 5 Saturdays, and 5 Sundays occurs nearly every year – in 2016 both January and July will have 5 Fridays, 5 Saturdays and 5 Sundays.
In fact, the calendar 823 years in the future is not even the same as the calendar of the month in the many versions of the hoax! For instance, August 2014 had 5 Fridays, 5 Saturdays, and 5 Sundays. 823 years later, August 2837 has 5 Saturdays, 5 Sundays and 5 Mondays.

The combinations of days that these messages or emails refer to are also not very special. This is because the first three weekdays of any 31-day month are repeated 5 times within that month. So, any month that has 31 days and begins on a Friday will have 5 Fridays, 5 Saturdays, and 5 Sundays. Similarly, any 31-day month that begins on a Tuesday, will have 5 Tuesdays, 5 Wednesdays and 5 Thursdays.

Here's a more mathematical reasoning as to why the 823 year claim is a myth:
Every year has 7 months with 31 days. Each of these months have at least 3 weekdays that happen 5 times during the month. For example, August 2014 has 5 Fridays, 5 Saturdays and 5 Sundays. October 2014 has 5 Wednesdays, 5 Thursdays and 5 Fridays.
There are 7 different days a month can start on. So on average, the combination of any 3 days occuring 5 times in a calendar month will happen once a year.
In any given year, there are only 6 such combinations of days. This is because 2 months with 31 days start on the same weekday. In a common or regular year, January and October have the same calendar, while in a leap year, January and July look the same.

#myth