Question 26

$$13$$ boys are sitting in a row in a theatre. After the intermission, they return and are seated such that either they occupy the same seat or the adjacent seat in such a way that it differs from the original arrangement. The number of ways this is possible is

Correct Answer: 376

Solution

Let $$S_n$$ denote the number of permitted rearrangements of $$n$$ boys when each boy can remain in the same seat or move to an adjacent seat. Then $$S_n=S_{n-1}+S_{n-2}$$, with $$S_2=2$$ and $$S_3=3$$. This gives $$S_{13}=377$$, and excluding the original unchanged arrangement leaves $$377-1=376$$.

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$$13$$ boys are sitting in a row in a theatre. After the intermission, they return and are seated such that either they occupy the same seat or the adjacent seat in such a way that it differs from the original arrangement. The number of ways this is possible is

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