Maximum number of tetrapeptides that can be prepared from the 10 essential amino acids is (a) 10,000 (b) 5000 (c) 2500 (d) 15000

Understanding the principles of permutations with repetition is key to solving a wide range of combinatorial problems. In problems like forming tetrapeptides that consist of a sequence of amino acids, each position in the sequence can be occupied by any one of the available amino acids, and the same amino acid can occur multiple times in the sequence.

Mathematically, the formula to calculate permutations with repetition is denoted as \( n^r \), where \( n \) represents the number of available different items and \( r \) signifies the number of positions to fill. In the context of our problem, with 10 essential amino acids and 4 positions in a tetrapeptide, we calculate the permutations as \( 10^4 \), which represents all the ways we can arrange these amino acids allowing for duplicates.

Our bodies need amino acids to synthesize proteins, which are critical to nearly every bodily function. Out of the twenty standard amino acids, there are nine that humans cannot synthesize, known as essential amino acids. These must be obtained through our diet.

These essential amino acids include Histidine, Isoleucine, Leucine, Lysine, Methionine, Phenylalanine, Threonine, Tryptophan, and Valine. Understanding these amino acids is pivotal in biochemistry and for anyone looking to solve problems related to protein synthesis or nutrition.

Peptides and proteins are fundamental components of cells that carry out important biological functions. A tetrapeptide is a molecule formed by linking together four amino acids with peptide bonds. The sequence and type of amino acids determine the structure and function of the peptide.

Proteins are comprised of one or more polypeptide chains, each folded into a specific three-dimensional structure. This structure is essential for the protein's function. Proteins can have four levels of structure - primary, secondary, tertiary, and quaternary. All these levels of structure influence the final shape and function of the protein in the biological system.

Students aiming to excel in the IIT JEE, one of India's most challenging engineering entrance exams, must thoroughly understand and practice a broad spectrum of concepts across Physics, Chemistry, and Mathematics. For Chemistry, a strong grasp of organic and inorganic chemistry, including the knowledge of amino acids and protein structure, is essential.

As potential candidates prepare for the exam, practicing combinatorial problems like calculating the number of tetrapeptides can sharpen their problem-solving skills. Furthermore, understanding the underlying principles of chemistry - from molecular interactions to chemical bonding - can provide a significant advantage in achieving success in the IIT JEE.