Unique calendars

Question

We have to make $3000$ unique calendars. There are unique in the sense that each calendar will have twelve designs (one for each month) in such a sequence that no two calendars are exactly identical.

Our intention is to create $x$ number of designs and print all the twelve months and dates on all of them and then sequence them in a unique manner.

Therefore, while we need to print $36000$ leaves ($3000 \times 12$), how many designs do we need to create $3000$ sets of $12$ that are all different from each other?

Answer 1

If you are happy for all $3000$ calendars to have the same designs, but in a different order (that's how I understood your question), then all you need is $12$ designs. The number of ways these can be ordered is $$12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$$ which equals $$479001600$$ which is way more than you need! In fact it is enough to have $3000$ different designs every year for almost $1600$ centuries!