How many different 5 letter sequences can be made using the letters A, B, C, D with repetition such that the sequence does not include the word BAD?
(a) 1024
(b) 976
(c) 48
(d) 678

Hint: We start solving the problem by recalling the fact that each position can be filled by 4 digits when repetition is allowed. We then find the total number of different 5 letter sequences that can be formed with the given letters when repetition is allowed. We then find the total number of sequences that can be formed by including the word BAD in it. We then subtract the sequences with word BAD in it from the total number of sequences to get the required answer.

Complete step by step answer:
According to the problem, we are asked to find the total number of different 5 letter sequences that can be made using the letters A, B, C, D with repetition such that the sequence does not include the word BAD.
Let us first find the total number of 5 letter sequences that can be formed with letters A, B, C, D.
We know that the 5 letters sequences will be of the form as shown below:

From the problem, it is said that repetition is allowed in each position of the sequence. So, each position can be filled in four ways.
So, the total number of sequences formed will be ${{4}^{5}}=1024$ ---(1).
Now, let us find the number of sequences that can be formed by including the word BAD in it.
Let us now write the possibilities of sequences with the word BAD in it.





We can see that in the three possibilities, we need to fill the remaining two places. We know that each place can be filled in 4 ways. So, we get the total number of sequences including the word BAD in it as $3\times 4\times 4=48$ ---(2).
Now, let us subtract the result obtained in equation (2) from equation (1) to get the sequences that do not include the word BAD.

So, the required number of sequences will be $1024-48=976$.

Note: Whenever we get this type of problem, we try to solve it indirectly by subtracting the unrequired cases from the total number of cases as there is a high chance of missing one or more cases while finding the sequences manually. We should check whether repetition is allowed before solving this type of problem. Similarly, we can expect a number of different 4 letter sequences that can be formed with repetition not allowed such that it doesn’t include word BAD.