How many permutations of the letters ABCDEFGH contain a) the...

How many permutations of the letters ABCDEFGH contain
a) the string ED?
b) the string CDE?
c) the strings BA and FGH?
d) the strings AB, DE, and GH?
e) the strings CAB and BED?
f ) the strings BCA and ABF?

Key Concepts

Permutation Counting

This concept involves determining the number of possible arrangements of a set of distinct items. In combinatorics, when all items are used, the total number of arrangements is computed as the factorial of the number of items, provided there are no restrictions.

Block (or Gluing) Method

This method is used when a particular sequence (or block) of items must appear together in the specified order within a permutation. The idea is to treat the entire block as a single unit, thereby reducing the number of objects to permute. This technique simplifies the counting process in problems with restrictions involving consecutive elements.

Counting with Constraints

Often in permutation problems, additional restrictions such as fixed substrings or multiple required blocks are introduced. This concept involves adapting standard permutation techniques to account for these constraints by reducing or modifying the problem (for example, by combining consecutive elements into a single unit) so that the restrictions are inherently satisfied.

Handling Overlapping Blocks

In cases where required substrings might overlap or interact, special attention is needed. This concept explains how overlapping conditions may affect the count when using the block method, potentially requiring adjustments or the use of additional techniques such as the inclusion?exclusion principle to avoid overcounting or undercounting.

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