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Find the number of all $20$-digit integers in which no two consecutive digits are the same.
in Discrete Mathematics by Expert | 19 views

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The first digit can not be $0$ and so for the first digit we have 9 options (from $1$ to  $9$). Now we have to choose the next digit in such a way that it is not the same as the previous one.  Thereofore on choosing the next digit we can have 9 options as well, (because we can choose any number from $0$ to $9$ except the previous one). Therefore, each of the remaining 19 digits can only have 9 options. Therefore there are $9\times 19^9$ numbers are available of the given type.
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