To calculate the number of different polypeptides that can be formed from n amino acids, you use the factorial function, denoted as n! (n factorial), which is the product of all positive integers from 1 up to n. Now, let's go through each specific number of amino acids you've asked about.
For 4 amino acids:
The number of different polypeptides is 4!, which is equal to:
4! = 4 × 3 × 2 × 1 = 24
So, 24 different polypeptides could be formed from 4 amino acids.
For 5 amino acids:
The number of different polypeptides is 5!, which is equal to:
5! = 5 × 4 × 3 × 2 × 1 = 120
Thus, 120 different polypeptides could be formed from 5 amino acids.
For 6 amino acids:
The number of different polypeptides is 6!, which is equal to:
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
Therefore, 720 different polypeptides could be formed from 6 amino acids.
For the 20 natural amino acids:
The number of different polypeptides is 20!, which is equal to:
20! = 20 × 19 × 18 × 17 × ... × 3 × 2 × 1
This value is quite large and would typically require a calculator or computer to compute accurately. Nonetheless, using a factorial function on a calculator, one could find the precise number. The value would indeed be a very large number reflecting the vast diversity possible in protein structures with the 20 natural amino acids.
A quick reference from a calculator or mathematical software shows that 20! is approximately equal to 2.4329 × 10^18, implying an extremely high number of potential different polypeptides.
