Final answer:
To find the number of ten-letter DNA sequences without consecutive identical letters, start with 4 options for the first letter, then 3 for each following letter, yielding 4 x 3^9 = 78,732 possible sequences.
Step-by-step explanation:
The question asks how many ten-letter DNA sequences do not have the same letter consecutively. Since there are four possible nucleotides (adenine (A), cytosine (C), guanine (G), and thymine (T)), the first letter of the sequence can be any of the four. After the first nucleotide has been chosen, the next nucleotide can only be one of the three remaining options to avoid consecutive repeats. This process is repeated for each subsequent nucleotide.
Step-by-step explanation:
- Choose the first nucleotide: There are 4 options (A, C, G, T).
- Choose the second nucleotide: Since the second cannot be the same as the first, there are 3 options left.
- For the third nucleotide and each nucleotide thereafter up to the tenth, the logic from step 2 applies, meaning there are again 3 options for each subsequent nucleotide.
Therefore, the total number of non-repeating ten-letter sequences is calculated as:
4 (choices for the first letter) x 3 (choices for the second letter) x 3 (choices for the third letter) x ... x 3 (choices for the tenth letter). That is, 4 x 39 = 4 x 19,683 = 78,732 possible non-repeating ten-letter DNA sequences.