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Consider a 5×5 array. In how many ways can we fill the array with X-s and O-s so that no two consecutive rows are identical?

So first one we can do $2^5$ ways. Second row in $2^5-1$, the same as 3,4,5 rows.

So final answer is $2^5*(2^5-1)^4$

Is it correct way of thinking?

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    It should be a product not sum, i.e. $2^5(2^5-1)^4$. – 2017-02-05
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    i think, due to rotational symmetry, it is overcounted – 2017-02-05
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    @AnuragA Yeah, sorry my bad. – 2017-02-05
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    @Kiran So how would you solve this task? – 2017-02-05
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    @sswwqqaa, i am also thinking in the same way. i am checking with a small example whether your answer is right. – 2017-02-05
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    @Kiran what do you mean by rotational symmetry? It is an array so the first row and fifth row cannot be thought of as consecutive. – 2017-02-05
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    for a , 2×2 array, due to a rotation of 90 degree, whether $\left[ \begin{array}{c c|c} X&O\\ O&X \end{array}\right] \left[ \begin{array}{c c|c} O&X\\ X&O \end{array}\right]$ are same? – 2017-02-05
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    I don't think that this interpretation holds in the present context. But I will let OP chime in on this. – 2017-02-05

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