umber of cyclic tripeptides formed with 22 amino acids AA and BB is:JEE Main 2023 (29 Jan Shift 1)A22B33C55D4

To calculate the number of cyclic tripeptides formed with 22 amino acids (AA and BB), we need to consider the following steps:

1. ### Break Down the Problem

We need to find the combinations of cyclic tripeptides formed from 22 amino acids. A tripeptide consists of three amino acids.

2. ### Relevant Concepts

For cyclic tripeptides, we can use the formula for combinations of n items taken r at a time, which is:

$C(n, r) = \frac{n!}{r!(n-r)!}$

Since order matters in peptides, we'll also need to consider the arrangements (permutations) of the selected amino acids.

3. ### Analysis and Detail

• We are interested in cyclic arrangements of tripeptides, which can be calculated by fixing one amino acid and arranging the others around it.

For tripeptides:

• We can select any 3 amino acids from the 22, and since cyclic arrangements are considered, there are $(3-1)!$ = 2 ways to arrange them after fixing one.

Using the combination formula, we need to choose 3 amino acids from 22:

$C(22, 3) = \frac{22!}{3!(22-3)!} = \frac{22 \times 21 \times 20}{3 \times 2 \times 1} = 1540$

Each selection can be arranged in 2 distinct cyclic ways, so we multiply by 2:

$1540 \times 2 = 3080$

4. ### Verify and Summarize

We calculated our combinations and arrangements. The formula used and the calculations were verified for accuracy.

Final Answer

The total number of cyclic tripeptides that can be formed with 22 amino acids is $3080$.