How many different sequences of length 6 are there in a three-letter alphabet? - briefly
In a three-letter alphabet, each position in a sequence of length 6 can be filled by any of the three letters. Therefore, the total number of different sequences is (3^6 = 729).
How many different sequences of length 6 are there in a three-letter alphabet? - in detail
To determine the number of distinct sequences of length 6 using an alphabet consisting of only three letters, we need to employ combinatorial principles. Let's delve into the details step by step.
Firstly, consider each position in the sequence as an independent choice from the three available letters. Since there are three options for the first letter, three options for the second, and so on up to the sixth letter, we can express the total number of possible sequences using exponentiation. Mathematically, this is represented as:
[ 3^6 ]
This calculation accounts for every possible combination where each position in the sequence can be filled with any of the three letters from the alphabet. Breaking down the calculation:
- For the first position, there are 3 choices (let's denote them as A, B, and C).
- Similarly, for the second position, there are again 3 choices.
- This pattern continues for each of the six positions in the sequence.
Therefore, to find the total number of distinct sequences, we multiply the number of choices for each position:
[ 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 ]
Performing the multiplication yields:
[ 3^6 = 729 ]
Thus, there are 729 different sequences of length 6 in a three-letter alphabet. This comprehensive approach ensures that every possible permutation and combination is accounted for, providing an accurate and exhaustive count of the distinct sequences.