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**Hint:**DNA is a double-stranded molecule that carries the genetic information of an organism. Deoxyribose sugar, a phosphate (phosphoric acid), and a nitrogen base together form a nucleotide. The deoxyribose sugar and the phosphate form the backbone of the DNA. Two polynucleotide chains that are helically coiled make up a DNA molecule.

**Complete answer:**

Nucleotides are energy-rich molecules. ATP or adenosine (adenine and deoxyribose sugar) triphosphate (three phosphate molecules) is one of the main molecules that aids in cellular metabolism. DNA nucleotides are formed by different combinations of the four nitrogen bases with the sugar and phosphate.

The four nitrogen bases are- adenine (A), guanine (G), thymine (T), and cytosine (C). Adenine and guanine are called purines. Thymine and cytosine are called pyrimidines. The DNA can form a double-stranded structure because the purines and pyrimidines can base pairs. Adenine always pairs with thymine and guanine will always pair with cytosine.

The position of the nitrogen base in the polynucleotide chain is very specific. Addition, deletion, or replacement of any base will change the DNA sequence. This change is called a mutation.

For this particular question, we are to assume that the nitrogen base can take up any random position in the polynucleotide chain. To find out the probability of getting a polynucleotide where the adjacent nitrogen bases read GAAT, we need to calculate the probability of getting a specific nitrogen base at each of the four positions. It is equally likely that all four nitrogen bases can take up any position out of the four.

The probability that guanine (G) occurs in the first position is \[\dfrac{1}{4}\]

The probability that adenine (A) occurs in the second position is $\dfrac{1}{4}$

The probability that adenine (A) occurs in the third position is $\dfrac{1}{4}$

The probability that thymine (T) occurs in the fourth position is $\dfrac{1}{4}$

The probability that the polynucleotide has a sequence that reads GAAT is ($\dfrac{1}{{4\,}}\, \times \,\dfrac{1}{4}\, \times \,\dfrac{1}{4}\, \times \,\dfrac{1}{4}$) that is equal to $\dfrac{1}{{256}}$.

This means that out of $256$ nitrogen bases only one will show GAAT nitrogen base sequence in the polynucleotide chain. In other words, out of the $256$ nitrogen bases synthesized only one will show the desired nitrogen base sequence.

**Hence, the correct answer is option (B).**

**Note:**Instead of deoxyribose sugar, the RNA nucleotide has a ribose sugar. As both sugar molecules are five carbon compounds they are called pentose sugar, in general. All nitrogen bases in the case of RNA molecules are the same as DNA molecules except for thymine. Uracil replaces thymine in the RNA molecule.

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