Question

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

Short answer

There are 120 different pentapeptides that can be formed using five different amino acids. This is calculated using the permutation formula \(P(5,5) = \frac{5!}{(5-5)!} = 120\).

Step-by-step solution

Identify number of different amino acids available

We have 5 different amino acids to arrange in a pentapeptide.

Understand permutations

Permutation is the arrangement of objects in a specific order. In this case, we want to find all possible arrangements of the 5 different amino acids. A formula for calculating permutations is given by: \[P(n,r) = \frac{n!}{(n-r)!}\] Where n is the total number of elements and r is the number of elements to be arranged. In this problem, we have 5 different amino acids and we need to arrange all the 5, so n = 5 and r = 5.

Apply the permutation formula

Now, let's apply the permutation formula to calculate the total number of unique arrangements that can be formed using the 5 different amino acids: \[P(5,5) = \frac{5!}{(5-5)!}\]

Solve for pentapeptides

We can now proceed to calculate the factorial values of 5! and 0!: \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\] \[0! = 1\] Now, plug these factorial values back into the permutation formula: \[P(5,5) = \frac{120}{1} = 120\] Hence, there are 120 different pentapeptides that can be formed using five different amino acids.