How many different sequences of length 6 are possible 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. Therefore, the total number of possible sequences is 3 raised to the power of 6, which equals 729.
How many different sequences of length 6 are possible in a three-letter alphabet (CAT)? - in detail
When considering sequences of length 6 composed of a three-letter alphabet (C, A, T), it is essential to understand the fundamental principles of combinatorial mathematics. Each position in the sequence can be independently filled with any of the three letters. This independence is crucial for calculating the total number of possible sequences.
To determine the number of different sequences, we must consider each position in the sequence separately. Since there are three choices (C, A, T) for each of the six positions, the total number of sequences can be calculated using the multiplication principle. This principle states that if there are ( n ) independent choices to be made, and each choice has a fixed number of options, the total number of possible sequences is the product of the number of options for each choice.
For a sequence of length 6, with three possible letters for each position, the calculation is straightforward:
- For the first position, there are 3 choices (C, A, T).
- For the second position, there are again 3 choices (C, A, T).
- This pattern continues for all six positions.
Thus, the total number of sequences is ( 3 \times 3 \times 3 \times 3 \times 3 \times 3 ). This can be simplified using exponential notation as ( 3^6 ).
Calculating ( 3^6 ) yields 729. Therefore, there are 729 different sequences of length 6 possible in a three-letter alphabet (C, A, T). This result is derived from the combinatorial principle of independent choices, where each position in the sequence is filled independently of the others, and the total number of sequences is the product of the number of choices for each position.
This detailed explanation illustrates the process of determining the number of sequences and the underlying mathematical principles involved. Understanding this process is vital for various applications in information theory, coding, and combinatorial problems.