Permutation question

Question

A mint prepares metallic calendars specifying month, dates and days in the form of monthly sheets (one plate for each month). how many types of calendars to serve for all Possibilities in futur years?

Answer 1

As an example to get you thinking, remember the decimal number system is a positional number system. So, with 10 digits (0-9) in 3 positions, we may construct 10*10*10 numbers (that is 1000, from 000-999).

Also, for a date of the form mm/dd, we have 12 possible months and 31 possible days, so that means 12*31 = 372 possible dates. Now, don't get confused -- we just don't use some of the possible mm/dd combinations in a real year [for example 2/30 is not used]. So, to determine the total number of mm/dd combinations that are actually needed, instead of 12*31, we add 31+28(or 29)+31+30+31+30+31+31+30+31+30+31 = 365(or 366). We could also group those into  7*31+4*30+1*28(or 29).

 

Now, if we wanted dates that look like mm/dd/yy for values of yy from 2017 to 2020 (3 regular years and 1 leap year), we would have 365+365+365+366 instead of 4*365.

O.K., the key to this problem is to determine the number of possible values for each variable.

Months:    13         [named January, February, February, March, …, December]
[note that February has a 28-day version and a 29-day version]

Day of week for day 1:    7        [positions on calendar are S, M, T,…, S]

Thus, 7*13= 91 plates cover all possibilities of months and starting day [and, eventually, we will use them all].