How many different symbol sequences of length 6 are there in a three-letter alphabet "cat"? - briefly
In a three-letter alphabet consisting of "c," "a," and "t," each position in a sequence of length 6 can be filled by any of the three letters. Consequently, the total number of different symbol sequences of length 6 is 3 raised to the power of 6, which equals 729.
How many different symbol sequences of length 6 are there in a three-letter alphabet "cat"? - in detail
To determine the number of different symbol sequences of length 6 in a three-letter alphabet consisting of the letters "c," "a," and "t," we need to understand the fundamental principles of combinatorics, specifically the concept of permutations with repetition.
A sequence of length 6 means that each position in the sequence can be filled by any of the three letters. Since there are three possible choices for each position, and there are six positions to fill, we can calculate the total number of sequences by multiplying the number of choices for each position.
Here is the detailed process:
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Identify the Alphabet and Sequence Length: The alphabet consists of three letters: "c," "a," and "t." The sequence length is 6.
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Understand the Combinatorial Principle: Each position in the sequence can be independently filled by any of the three letters. This is a classic example of permutations with repetition, where the number of choices for each position remains constant.
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Calculate the Total Number of Sequences: Since there are three choices for each of the six positions, the total number of different sequences can be calculated using the formula for permutations with repetition:
[ n^k ]
where ( n ) is the number of choices for each position (in this case, 3), and ( k ) is the length of the sequence (in this case, 6).
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Substitute the Values: Substitute ( n = 3 ) and ( k = 6 ) into the formula:
[ 3^6 ]
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Compute the Result: Calculate ( 3^6 ):
[ 3^6 = 729 ]
Therefore, there are 729 different symbol sequences of length 6 that can be formed using the three-letter alphabet "cat". Each sequence is unique and consists of a combination of the letters "c," "a," and "t" arranged in various orders over six positions. This calculation illustrates the power of combinatorial principles in determining the number of possible sequences in a given alphabet and sequence length.