For cyclic peptides:
• Count unique sequences considering the cyclic structure to avoid duplicates.
• Use the number of amino acids and their combinations to determine possible arrangements.
4
2
1. Cyclic Tripeptides:
- A cyclic tripeptide is a peptide consisting of three amino acids arranged in a cyclic structure without any free ends.
- Given two amino acids, A and B, the tripeptide can have different arrangements depending on the sequence of A and B.
2. Possible Arrangements:
- The cyclic nature of the tripeptide eliminates linear sequence redundancy.
- The distinct arrangements are:
1. A-A-A
2. A-A-B
3. A-B-A
4. A-B-B
Total: 4 unique cyclic tripeptides.
If $ \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p $, then $ 96 \log_e p $ is equal to _______
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: