Problem 596

【Example 1】Our country is the earliest in the world to discuss permutations with repetition, and the materials collected in the *I Ching* can be traced back to the 7th century BC. In the *I Ching*, there are two symbols, called "yang" and "yin." The *I Ching* discusses all possible cases formed by these two symbols in 3-character and 6-character sequences. The 3-character sequences are:
yang yang yang, yang yang yin, yang yin yang, yang yin yin,
yin yang yang, yin yang yin, yin yin yang, yin yin yin.

This is precisely the result of all possible selections of 3 repetitions from the set {yang, yin}, or the permutations with repetition of selecting 3 from yang and yin. Since the number of cases is $2 \times 2 \times 2=2^{3}=8$, the *I Ching* refers to them as the "Eight Trigrams." Correspondingly, there are $2^{6}=$ 64 different cases for 6-character sequences, known as the "Sixty-Four Hexagrams."