Final answer:
The correct number of permutations of the letters abcdefgh that contain the string cde is given by considering cde as one entity, and thus there are 6 entities to permute. The correct answer is 6×6!, which is option b).
Step-by-step explanation:
The question asks for the number of permutations of the letters abcdefgh that contain the string cde. To solve this, consider the string cde as a single entity. This leaves us with five more entities to arrange: a, b, f, g, h, and the entity 'cde'. Therefore, we have a total of 6 entities to arrange. The number of permutations of these six entities is 6! (6 factorial), which is 6×5×4×3×2×1.
We do not need to consider the internal arrangement of the string cde since it must remain in that order to satisfy the condition. Thus, the answer to the question is simply the permutations of the six entities, which is 6×6!.
Now, 6! is the number of permutations of 6 different entities, which is 720. Multiplying this by 6, we get a total of 4320 permutations. Therefore, the correct option is b) 6×6!, which gives us the answer.