Understanding Disulfide Bond Formation
The question asks for the number of ways to form 5 intramolecular disulfide bonds using 10 cysteine residues. A disulfide bond forms between two cysteine residues. Since we have 10 cysteines and need 5 bonds, all cysteines are involved in forming pairs.
Calculating Possible Pairings
This is a combinatorial problem of pairing up 10 distinct items (cysteine residues). We can calculate this using the formula for partitioning $2n$ items into $n$ pairs, which is given by the double factorial $(2n-1)!!$.
In this case, $N = 10$ cysteine residues, so $n = N/2 = 5$ pairs (disulfide bonds).
The number of ways is calculated as $(2 \times 5 - 1)!! = 9!!$.
Double Factorial Calculation
The double factorial $9!!$ is calculated as:
$ 9!! = 9 \times 7 \times 5 \times 3 \times 1 $Performing the multiplication:
$ 9 \times 7 = 63 $ $ 63 \times 5 = 315 $ $ 315 \times 3 = 945 $ $ 945 \times 1 = 945 $Therefore, there are 945 different possible ways to form five intramolecular disulfide bonds with ten cysteine residues.
Alternative Combinatorial Calculation
Alternatively, we can choose pairs sequentially and account for overcounting:
- Choose the first pair from 10 cysteines: $\binom{10}{2}$ ways.
- Choose the second pair from the remaining 8: $\binom{8}{2}$ ways.
- Choose the third pair from the remaining 6: $\binom{6}{2}$ ways.
- Choose the fourth pair from the remaining 4: $\binom{4}{2}$ ways.
- Choose the fifth pair from the remaining 2: $\binom{2}{2}$ ways.
The initial calculation is $\binom{10}{2} \times \binom{8}{2} \times \binom{6}{2} \times \binom{4}{2} \times \binom{2}{2}$.
Calculating the values:
$ \frac{10 \times 9}{2} \times \frac{8 \times 7}{2} \times \frac{6 \times 5}{2} \times \frac{4 \times 3}{2} \times \frac{2 \times 1}{2} = 45 \times 28 \times 15 \times 6 \times 1 = 113,400 $Since the order of forming the 5 pairs does not matter, we divide by $5!$ (the number of ways to order the pairs):
$ \frac{113,400}{5!} = \frac{113,400}{120} = 945 $Both methods confirm that there are 945 possible ways.
Important Questions from Combinations
How many ways are there to pack six copies of the same book into four identical boxes, where a box can contain as many as six books ?
Let us assume a person climbing the stairs can take one stair or two stairs at a time. How many ways can this person climb a flight of eight stairs?
There are 9 different rock lizard species, each with a unique colour. Lizards of 2 different species sit on a rock at any given time. The number of possible colour combinations of rock lizards seen together on a rock is _________
(Answer in integer)- There are nine species of Impatiens (balsams) found in laterite plateaus of the northern Western Ghats, each with a distinct colour. If a plateau has exactly 6 species, then the number of possible colour combinations in the plateau is __________. (Answer in integer)
- A set of 4 parallel lines intersect with another set of 5 parallel lines. How many parallelograms are formed?
