Question

A mint prepares metallic calendars specifying months, dates, and days in the form of monthly sheets (one plate for each month). How many types of February calendars should it prepare to serve for all the possibilities in the future years?

Answer 1

Hint: For solving this question, first we will see the concept of the fundamental principle of multiplication. After that, we will find the number of different types of month February is possible as per the type of year (leap year and non-leap year) and the first day of February. Then, we will apply the fundamental principle of multiplication to get the final answer.

Complete step-by-step answer:
Given:
It is given that a mint prepares metallic calendars specifying months, dates, and days in the form of monthly sheets (one plate for each month). And we have to find the total number of types of February calendars should it prepare to serve for all the possibilities in the future years.Now, we can say that the total number of different calendars required will be equal to the number of different types of month February are possible.
Now, before we proceed we should know the following important concept and formulas:
Fundamental Principle of Multiplication:
If there are two jobs such that one of them can be completed in $m$ ways, and when it has been completed in any of these $m$ ways, the second job can be completed in $n$ ways. Then, two jobs in succession can be completed in $m\times n$ ways.
Now, for the month February if the type of year (leap year and non-leap year) and the first day of the month (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday) is the same. Then, those calendars can be used again.
Now, as we know that, for a leap year there will be 29 days in February and 28 days otherwise. Moreover, there will be seven choices (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday) for the first day of February.
Number of choices as per the type of year (leap year and non-leap year) $=m=2$ choices.
Number of choices as per the first day of the month (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday)$=n=7$ choices.
Now, from the above discussion, on the number of choices available and fundamental principle of multiplication, we can say that the total number of calendars required will be equal to $m\times n=2\times 7=14$ .
Thus, he should prepare $14$ types of February calendars to serve for all the possibilities in the future years.

Note:Here, the student should first understand what is asked in the question and then proceed with the right approach to get the correct answer quickly. Moreover, we should apply the fundamental principle of counting very carefully and don’t forget to include leap year while solving the question as we have to find the answer for February.