Question

How many different tetrapeptides can be made under the following conditions? (a) The tetrapeptide contains one unit each of Asp, Glu, Pro, and Phe. (b) All 20 amino acids can be used, but each only once.

Short answer

Answer: Under condition (a), 24 different tetrapeptides can be formed. Under condition (b), 116,280 different tetrapeptides can be formed.

Step-by-step solution

Understanding permutations

Permutations are an arrangement of objects in a specific order. In this exercise, we are looking for the number of ways to arrange the amino acids to form a tetrapeptide. The number of permutations can be found using the formula: Permutations (n, r) = n! / (n - r)! where "n" is the number of objects to choose from, "r" is the number of objects to arrange, and "n!" represents the factorial of n (i.e., the product of all positive integers up to n).

Calculate the permutations for part (a)

In part (a), we have 4 distinct amino acids {Asp, Glu, Pro, Phe}, and we need to form a tetrapeptide using each of them once, which means that n = 4 and r = 4. By using the permutation formula: Permutations (4, 4) = 4! / (4 - 4)! = 4! / 0! = 4! / 1 = 4 x 3 x 2 x 1 = 24 So, there are 24 different tetrapeptides that can be formed under condition (a).

Calculate the permutations for part (b)

In part (b), we have 20 distinct amino acids, and we need to form a tetrapeptide using 4 of them, with each amino acid being used only once. This means that n = 20 and r = 4. Using the permutation formula: Permutations (20, 4) = 20! / (20 - 4)! = 20! / 16! = (20 x 19 x 18 x 17) = 116280 So, there are 116,280 different tetrapeptides that can be formed under condition (b).