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 A₁, there’s one choice to arrange it. Next, take A₂. It can be squeezed into A₁ or placed next to it, giving a total of 2 choices. Now, we have 3 choices for A₃: directly contained by A₁, directly contained by A₂, or standalone. This is true regardless of the choice made with the placement of A₂. Following similar logic, A₄ 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 A₆, when A₁– A₅ are already in place? Consider the following.

• There are at least 6 options:
  • A6 can only be directly contained in A₁, A₂, A₃, A₄, A₅, or nothing. These are all different cases.
• There are at most 6 options:
  • If there is the 7th option, then what envelope is outside of A₆? It must be one of A₁-A₅ or nothing at all. We’ve already considered these cases above, so in fact there cannot be any new cases.

Hence, in total we have 6! = 720 options.