Competition problem — Combinatorics

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Combinatorics Difficulty 5.1 AIME, harder Prove it United States

Problem:

In how many ways can $8$ people be arranged in a line if Alice and Bob must be next to each other, and Carol must be somewhere behind Dan?

Solution

Solution:

Let us place Alice and Bob as a single person; there are then $7! = 5040$ different arrangements. Alice can be in front of Bob or vice versa, multiplying the number of possibilities by $2$ , but Carol is behind Dan in exactly half of those, so that the answer is just $5040$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.