How many different symbolic sequences of length 5 are there in a four-letter alphabet "cat"? - briefly
In a four-letter alphabet such as "cat," each position in a sequence of length 5 can be filled by any of the 4 available letters. Therefore, there are (4^5 = 1024) different symbolic sequences of length 5.
How many different symbolic sequences of length 5 are there in a four-letter alphabet "cat"? - in detail
In the context of combinatorics, determining the number of distinct symbolic sequences of a given length within a specific alphabet is a fundamental problem. When dealing with an alphabet consisting of four letters—specifically, the letters in the word "cat"—we need to calculate the total number of possible sequences of length 5 that can be formed using these four symbols.
To begin, let's identify our alphabet: {c, a, t}. Despite the word "cat" consisting of only three unique letters, we must consider that each position in the sequence can be filled with any of these three letters. This means we are dealing with sequences that can include repetitions of the same letter.
The key to solving this problem lies in understanding the principle of combinatorial multiplicity. For each position in the sequence, there are exactly three choices (c, a, or t). Since the sequence is of length 5, and each of the five positions can be independently filled with any of the three letters, we use the formula for combinations with repetition:
[ n^k ]
where ( n ) is the number of available symbols in the alphabet, and ( k ) is the desired length of the sequence. In our case, ( n = 3 ) (the letters c, a, t) and ( k = 5 ) (the length of the sequence). Applying this formula:
[ 3^5 ]
This calculation yields:
[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 ]
Therefore, there are 243 different symbolic sequences of length 5 that can be formed using the letters in the alphabet "cat". This result underscores the vast combinatorial possibilities even with a relatively small set of symbols and sequence lengths.