Play with words.

Number of total words - Number of “BAD” words: Number of total words is $4^5 = 1024$ . Number of BAD words look like BADxx, xBADx, xxBAD (where x is any letter). There are 16 choices for each time of BAD word; for a total of 48 BAD words. Thus, there are 976 good sequences.

For each of the five places for a letter there are 4 choices of a letter to fill that space (A, B, C or D). This means that there are 4 x 4 x 4 x 4 x 4 = $4^ 5$ arrangements of the letters. However, we need to subtract the number of arrangements containing the word BAD. Either the word BAD is found at the start of the arrangement with two letters afterwards, at the end of the arrangement with two letters before or in the centre with one letter before and one afterwards. For each of these arrangements, there are 4 choices of a letter for each of the two remaining spaces so there are $4^2$ arrangements containing BAD at the start, $4^2$ with BAD at the end and $4^2$ with BAD in the centre. So in total there are $4^5$ - (3 x $4^2$ ) = $\boxed{976}$ arrangements of the letters that fit the criteria given.

all 5 leter = 4 4 4 4 4 = 1024 bad at first = BAD * = 1 4 4 = 16 bad at second = *BAD = 4 1 4 = 16 bad at third = *BAD = 4 4 1 = 16 sequence does not include the word BAD = 1024 - 3 16 = 976

Number of ways to choose an ordered sequence of 5 letters from a set of 4 letters {A,B,C,D} with repetition is : $4^5 = 1024$ (4 choices for the first letter, 4 choices for the second, etc).

Now, these 1024 arrangements evidently include the sequences containing the word BAD. So, all that is now required to do, is to find all the sequences containing this word and subsequently subtract them from 1024.

So, the sequence we are looking for contain the word BAD. These 3 letters are either the first 3 letters or the 2nd, 3rd and 4th letter or the last 3 letters in our sequence. The remaining 2 letters are allowed to be any other letter.

Hence, number of choices is : $3*4^2 = 48$

So, the answer is : $1024-48 = \boxed{976}$