Problem 90 - Chemistry An Atoms First Approach

Question

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

Step-by-Step Solution

Answer

There are 120 different pentapeptides that can be formed using the given five amino acids. This is calculated using the permutation formula \(P(n, r) = \frac{n!}{(n-r)!}\), where n is the number of amino acids and r is the number of positions to fill. In this case, \(P(5, 5) = \frac{5!}{(5-5)!} = \frac{120}{1} = 120\).

Step 1: Define the problem

We need to find the number of ways to arrange five different amino acids to form pentapeptides. Since a pentapeptide has five positions for the amino acids, we will be arranging the five amino acids to fill these positions without repetition.

Step 2: Permutation formula

The permutation formula, denoted by P(n, r), is a way to find the number of ways to arrange 'n' items in 'r' positions without repetition. The formula is: P(n, r) = n!/(n-r)! In our case, we have 5 different amino acids and 5 positions to fill, so n=5 and r=5.

Step 3: Calculate the factorial of n and n-r

First, we need to calculate the factorial of 'n' (5) and 'n-r' (0): - 5! = 5 × 4 × 3 × 2 × 1 = 120. - 0! is defined as 1.

Step 4: Calculate the permutations using the formula

Now, we can calculate P(5, 5) using the formula. P(5, 5) = 120/1 = 120

Step 5: Write the final answer

There are 120 different pentapeptides that can be formed using the given five amino acids.