How many different symbol sequences of length 5 are possible in a three-letter alphabet cat? - briefly
A three-letter alphabet "cat" consists of three distinct symbols: 'c', 'a', and 't'. For sequences of length 5, each position in the sequence can be filled by any of the three letters. Thus, the total number of different symbol sequences possible is 3^5, which equals 243.
How many different symbol sequences of length 5 are possible in a three-letter alphabet cat? - in detail
To determine the number of different symbol sequences of length 5 possible in a three-letter alphabet consisting of the letters "c", "a", and "t", we need to understand the principles of combinatorial mathematics, specifically permutations with repetition.
A sequence of length 5 means each position in the sequence can be filled by any of the three letters. Since the order of the letters matters and repetition is allowed, we are dealing with permutations with repetition.
The formula for permutations with repetition is given by ( n^r ), where ( n ) is the number of different symbols available, and ( r ) is the length of the sequence. In this case, ( n = 3 ) (the letters "c", "a", and "t") and ( r = 5 ) (the length of the sequence).
To calculate the total number of sequences, we raise the number of symbols to the power of the sequence length:
[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 ]
Performing the multiplication step-by-step:
[ 3 \times 3 = 9 ] [ 9 \times 3 = 27 ] [ 27 \times 3 = 81 ] [ 81 \times 3 = 243 ]
Therefore, the total number of different symbol sequences of length 5 possible in a three-letter alphabet "cat" is 243.
To illustrate, consider the possible sequences starting with each letter:
-
Sequences starting with "c":
- "ccccc"
- "cccca"
- "cccat"
- "ccac..."
- "ccata"
- ...
- "cttac"
-
Sequences starting with "a":
- "aaaaa"
- "aaaac"
- "aaaat"
- "aacac"
- "aacat"
- ...
- "attac"
-
Sequences starting with "t":
- "ttttt"
- "ttttc"
- "tttta"
- "tttac"
- "tttat"
- ...
- "tcctt"
Each of these groups will have 81 sequences, confirming that the total number of sequences is indeed 243.
This detailed analysis shows that the combinatorial approach is crucial for solving such problems, ensuring that all possible sequences are accounted for systematically.