Problem 60 - A Complete Resource Book in Chemistry for JEE Main

Question

The number of tripeptides formed by 3 different amino acids are? (a) 3 (b) 4 (c) 5 (d) 6

Step-by-Step Solution

Answer

The number of tripeptides formed is 6, so the answer is (d) 6.

Step 1: Understanding the Question

We need to calculate the total number of tripeptides that can be formed using 3 different amino acids. A tripeptide consists of three different amino acids arranged in various sequences.

Step 2: Calculating Arrangements

The number of different arrangements of 3 amino acids can be found by calculating the permutations of 3 distinct items, which is given by the factorial of 3, denoted as 3!. Calculate this using the formula for permutations.

Step 3: Applying the Permutation Formula

The permutation formula for arranging 3 different amino acids is given by the factorial of 3: \[ 3! = 3 \times 2 \times 1 = 6 \]

Step 4: Verification of Calculations

Verify that 3! correctly represents the number of unique tripeptide sequences by confirming the factorial computation. Since each amino acid can occupy any of the three positions in the tripeptide, the calculations are correct.

Key Concepts

Understanding Amino Acids

Amino acids are the building blocks of proteins. They are organic compounds that combine to form peptides, which then join to make proteins. Each amino acid consists of:

• a central carbon atom (also known as the alpha carbon)
• an amino group (-NH₂)
• a carboxyl group (-COOH)
• a hydrogen atom
• an R-group or side chain, which is unique to each amino acid

The diversity of proteins arises from the unique sequences and structures these amino acids form. There are 20 standard amino acids, each distinctive by its side chain. When these acids link together, they form peptides or proteins. This linkage is crucial because it allows the formation of different structures with various functions in the body. In the context of this problem, three different amino acids are arranged to form a tripeptide.

Tripeptides Explained

Tripeptides are specific types of peptides consisting of three amino acids linked by peptide bonds. These combinations can take numerous forms depending on the sequence of the amino acids. It’s essential to note that:

• The sequence of amino acids in a tripeptide matters. For example, ABC is different from BAC.
• Trippeptides demonstrate how amino acid sequences can influence their functional properties.
• Peptides, including tripeptides, play a significant role in biological processes such as enzymatic activity and cellular signaling.

Creating these sequences involves understanding permutations, which are arrangements where the order is important. For three different amino acids, each may be placed first, second, or third, leading to multiple potential combinations.

The Role of Factorials in Permutations

When determining the number of tripeptide sequences with three distinct amino acids, the concept of factorials is crucial. A factorial, denoted by an exclamation mark (!), is the product of an integer and all the integers below it.

• For example, 3! (read as "three factorial") is calculated as follows:\[ 3! = 3 \times 2 \times 1 \]
• Thus, 3! = 6 indicates there are six possible arrangements of three distinct items.

In the context of this exercise:

• Each amino acid can take one of the three positions, leading to a scenario where the permutation of three amino acids is calculated as 3!.
• This calculation means that, ultimately, there are six different ways to arrange these three amino acids into a tripeptide.

This simple application of factorials showcases how order and arrangement play a vital role in forming structures from amino acids, reinforcing the importance of permutations in combinatorial settings.