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A permutation and combination problem in high school mathematics
In the sequence of 15 letters arranged by letters A and B, how many of them meet the following conditions?

The letters AA appear five times in a row, AB, BA and BB appear three times each.

Solution:

Starting from the first item, suppose that each paragraph consisting of a is A 1, A2, A3, and

The paragraphs composed of b are B 1, B2, b3, ...

The arrangement shapes that meet the conditions can only be such combinations.

A 1 B 1 A2 B2 A3 B3 a4 ①

Or b1a1b2a2b3b3b4 ②.

Each segment contains at least 1 A (or B), occupying 7 bits and requiring 8 bits.

How to allocate the remaining five A's and three B's to form five AA's and three BB's?

If a paragraph consists of (5+ 1) A, then only five AA can be formed in this paragraph;

The remaining five A's and three B's can only be allocated in this way.

For ①, put A with C (8,5) = 56 (species), put B with C (5,3) =10 (species), * * * 56× 10 = 560 (species).

For ②, put A with c (7,5) = 21(species), put B with c (6,3) = 20 (species), * * * 2 1× 20 = 420 (species).

Therefore, the arrangement that meets the conditions is always * * * 560+420 = 980 (species).

This topic is too difficult.