Competition problem — Combinatorics

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Combinatorics Difficulty 2.8 Junior Find the answer

Five people are arranged in a row, where person A is not next to person B, and person A is also not next to person C. The number of different arrangements is

Pick one

Solution

According to the problem, we can use the indirect method to solve it.
First, calculate the total number of permutations of five people, which is $A_5^5$ .
The arrangements that do not meet the conditions are those where A is adjacent to B, and A is adjacent to C.
The number of arrangements where A is adjacent to B is $A_2^2A_4^4$ ,
and the number of arrangements where A is adjacent to C is $A_2^2A_4^4$ .
There is some overlap in these two groups of numbers that needs to be subtracted.
Therefore, the number of different arrangements where A is not adjacent to B, and A is also not adjacent to C is $A_5^5 - 2A_2^2A_4^4 + A_2^2A_3^3 = 36$ .
Hence, the correct answer is $\boxed{C}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.