**Tee Jet Whaw (Imperial Maths Competition 2020 Team Round)**

There are a total of 6 envelopes of different sizes. An envelope can contain one or more smaller envelopes. How many ways are there to pack the 6 envelopes into a parcel?

*Scroll down for solution!*

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### Solution

Don’t break into cases! Let A_{n} be the n-th largest envelope.

Starting from A_{1}, there’s one choice to arrange it. Next, take A_{2}. It can be squeezed into A_{1} or placed next to it, giving a total of 2 choices. Now, we have 3 choices for A_{3}: directly contained by A_{1}, directly contained by A_{2}, or standalone. This is true regardless of the choice made with the placement of A_{2}. Following similar logic, A_{4} has 4 options.

As we proceed, things start to get a little complicated… For example, how do we persuade ourselves that there are 6 options of where to place A_{6}, when A_{1}– A_{5} are already in place? Consider the following.

- There are at least 6 options:
- A6 can only be directly contained in A
_{1}, A_{2}, A_{3}, A_{4}, A_{5}, or nothing. These are all different cases.

- A6 can only be directly contained in A
- There are at most 6 options:
- If there is the 7th option, then what envelope is outside of A
_{6}? It must be one of A_{1}-A_{5}or nothing at all. We’ve already considered these cases above, so in fact there cannot be any new cases.

- If there is the 7th option, then what envelope is outside of A

H**ence, in total we have 6! = 720 options.**