How many strings of length 8 formed using letters from $\{a,b,c,d\}$ contain exactly one pair of adjacent letters that are the same?
Can any kind soul give some hint?
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- $\begingroup$ @user1202487, PLEASE stop this wholesale editing of old posts. Each time you make an edit, you bump a newer question off the front page. PLEASE STOP! $\endgroup$Gerry Myerson– Gerry Myerson2023-07-25 03:18:00 +00:00Commented Jul 25, 2023 at 3:18
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2 Answers
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Hint:
How many strings of length $8$ using those letters have the same first and second letters and have no other adjacent letters that are the same?
How many strings of length $8$ using those letters have the same second and third letters and have no other adjacent letters that are the same?
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First count the number of strings of length $7$ with no adjacent letters the same. We have four choices for the first letter, and three for each subsequent letter, so this is $4\times 3^6$.
Now choose one of these seven letters to be the repeated letter. So we get
$$7\times 4\times 3^6=20412$$
