2. Clever Dusya arranges six cheat sheets in four secret pockets so that the 1st and 2nd cheat sheets end up in the same pocket, the 4th and 5th cheat sheets also end up in the same pocket, but not in the same pocket as the 1st. The others can be placed anywhere, but only one pocket can remain empty (or all can be filled). In how many different ways can this be done?

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# Solution

Let's reason constructively and straightforwardly. Place the 1st and 2nd cheat sheets in any pocket. This can be done in 4 ways. Now place the 4th and 5th cheat sheets in any free pocket. This can be done in 3 ways. In total, there are 12 ways. There remain two cheat sheets (3rd and 6th) and two free pockets. There are two possible scenarios. If there are no empty pockets, there are two ways to arrange the remaining cheat sheets. If one of the two pockets is empty, the remaining cheat sheets can be placed in the three non-empty pockets in three different ways.

In total, this results in \(12 \cdot (2 + 2 \cdot (2 + 3)) = 144\) ways.

Answer: 144 ways.