Question

How many different pentapeptides can be formed using five different amino acids?

Short answer

Using five different amino acids, a total of \(120\) different pentapeptides can be formed. This is calculated using the permutation formula, where \(P(5, 5) = \frac{5!}{(5-5)!} = \frac{120}{1} = 120\).

Step-by-step solution

Understand the problem and required elements

Here, we have 5 different amino acids, and we need to create pentapeptides which consist of 5 amino acids. So, we are looking for the number of ways in which we can arrange these 5 amino acids.

Calculate the permutation

To calculate the permutation of n elements taken r at a time, we use the formula: \[P(n, r) = \frac{n!}{(n-r)!}\] In this problem, we have n = 5 (five different amino acids) and r = 5 (since each pentapeptide has 5 amino acids), so the formula becomes: \[P(5, 5) = \frac{5!}{(5-5)!}\]

Calculate the factorial values

Next, we need to calculate the factorial values of 5! and 0!. Factorial of a number (n!) is the product of all the numbers from 1 to n (n inclusive). - 5! = 5 × 4 × 3 × 2 × 1 = 120 - 0! = 1 (by definition, the factorial of 0 is always 1)

Calculate the number of pentapeptides

Now substitute the factorial values into the permutation formula: \[P(5, 5) = \frac{120}{1} = 120\]

Interpret the result

The result we obtained, 120, means that there are 120 different pentapeptides that can be formed by combining the 5 different amino acids.