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Question: A science communicator is creating a series of videos on permutations and wants to demonstrate all possible arrangements of 7 distinct science concepts, where 2 specific concepts must always appear adjacent to each other. How many such arrangements are there?
Solution: Treat the 2 specific concepts that must be adjacent as a single block. This reduces the problem to arranging 6 items (the 5 other concepts and the 1 block). These 6 items can be arranged in $6!$ ways. Within the block, the 2 concepts can be arranged in $2!$ ways. Therefore, the total number of arrangements is:
Now subtract to find the number of codes with at least two consecutive identical digits:

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