A four digit number is called a doublet if any of its digit is the same as only one neighbor. (atleast one digit is same as exactly one neighbour) (For example, 1221 is a doublet but 1222 is not), Number of such doublets are

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$2259$

$2268$

$2277$

$2349$

answer is B.

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Detailed Solution

Case — 1 abbc
We can choose a in 9 ways, b in 9 ways, and c in 9 ways. Hence, there are $9^{3}$ possibilities for this case.
Case — 2 bbac

There are 9 possibilities for b, and 9 possibilities for each a and c. Once again, we have that there are $9^{3}$ possibilities for this case.
Case — 3 acbb
Once again, there are 9 possibilities for a and c, and 9 possibilities for b. hence. these area $9^{3}$ possibilities for this case.
Case — 4 aabb
We have 9 possibilities for a and 9 possibilities for b. We thus have 81 possibilities for this case. We can now sum up our answers, to get a total of ${3.9}^{3} + 81 = 2268$ total ways.