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Chapter 6: Problem 18

How many 5-element DNA sequences a) end with A? b) start with T and end with G? c) contain only A and T? d) do not contain C?

Short Answer

Expert verified
a) 256 sequences; b) 64 sequences; c) 32 sequences; d) 243 sequences.

Step by step solution

01

Total possible sequences

For a DNA sequence of 5 elements, each position can be one of 4 nucleotides (A, T, C, G). Hence, the total number of 5-element DNA sequences is \[4^5\].
02

Sequences ending with A

To find sequences that end with A, fix the last element as A. The remaining 4 positions can still be any of the 4 nucleotides. Thus, there are \[4^4\] sequences that end with A.
03

Sequences starting with T and ending with G

Fix the first element as T and the last element as G. The 3 middle positions can still be any of the 4 nucleotides. Therefore, the number of such sequences is \[4^3\].
04

Sequences containing only A and T

If the sequence can only contain A and T, each position has 2 choices. Hence, the number of these sequences is \[2^5\].
05

Sequences not containing C

To ensure a sequence doesn鈥檛 contain C, each position has 3 choices (A, T, G). Thus, there are \[3^5\] sequences that do not contain C.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorial Analysis
Combinatorial analysis is a branch of mathematics focused on counting, arrangement, and combination of objects. In the context of DNA sequences, combinatorial analysis helps solve problems like how many different sequences can be formed under certain conditions. For example, when counting how many DNA sequences end with 'A', we apply combinatorial principles to determine the possible configurations.
Nucleotide Sequences
Nucleotide sequences are strings made up of nucleotides, which are the building blocks of DNA. There are four types of nucleotides in DNA: Adenine (A), Thymine (T), Cytosine (C), and Guanine (G). When we look at sequences of a specific length, like 5-element sequences in our exercise, nucleotide combinations form unique strings. For instance, in a 5-element sequence, each position can be any of the four nucleotides, producing and possible total sequences.
Probability in Sequences
Probability in sequences refers to the likelihood of a particular nucleotide arrangement appearing. This is crucial in calculating the number of specific DNA sequences. For example, to find how many sequences start with 'T' and end with 'G', we fix these positions and calculate the probabilities for the remaining positions. Thus, we apply probability principles to count how likely certain sequence patterns are to appear.
Discrete Mathematics
Discrete mathematics deals with distinct and separate values. In the context of DNA sequences, this involves analyzing sets of distinct sequences formed by specific rules. For example, if we want sequences that do not contain the nucleotide 'C', we are working within a subset of sequences created by excluding 'C' from choices at each position. Discrete mathematics provides the tools and methods necessary to count and analyze such specific sets of sequences.

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