please explain
How many ways are there to arrange 3 molecules on a symmetric polymer with 8 identical sites?
Ah, the wonders of molecular arrangements! Let me put on my lab coat and clown wig to give you an entertaining explanation.
Now, on a symmetric polymer with 8 identical sites, we want to arrange 3 molecules. It’s like organizing a party with a limited guest list!
To calculate the number of arrangements, we can use a formula known as the combination formula. It goes like this:
C(n, k) = n! / (k! * (n - k)!)
Here, "n" represents the total number of sites (8 in this case) and "k" is the number of molecules we want to arrange (3).
Plugging in the numbers, we have:
C(8, 3) = 8! / (3! * (8 - 3)!)
Simplifying further:
C(8, 3) = 8! / (3! * 5!)
By performing the math magic:
8! = 8 * 7 * 6 * 5! (since 8! = 8 * 7 * 6!)
The 5! in the denominator also cancels out with 5! in the numerator.
At last, we get:
C(8, 3) = (8 * 7 * 6) / (3 * 2 * 1)
C(8, 3) = 8 * 7 = 56
So, there are 56 hilarious ways to arrange 3 molecules on a symmetric polymer with 8 identical sites. That's a lot of microscopic party options!
To find the number of ways to arrange 3 molecules on a symmetric polymer with 8 identical sites, we can use the concept of combinations.
We have 8 identical sites for the molecules to be placed on. Since the polymer is symmetric, we can divide it into two halves, each containing 4 sites.
Now, we need to determine in how many ways we can distribute the 3 molecules between these two halves.
We can represent this using combination notation, where "n" represents the total number of sites and "r" represents the number of molecules to be distributed.
The formula for combinations is given by:
C(n, r) = n! / (r!(n-r)!)
In this case, the total number of sites (n) is 4, and the number of molecules (r) is 3.
Plugging in these values, we have:
C(4, 3) = 4! / (3!(4-3)!)
Simplifying this expression, we get:
C(4, 3) = 4! / (3! * 1!)
Evaluating the factorials:
4! = 4 * 3 * 2 * 1 = 24
3! = 3 * 2 * 1 = 6
1! = 1
Substituting these values back into the equation, we have:
C(4, 3) = 24 / (6 * 1)
Simplifying further, we get:
C(4, 3) = 24 / 6 = 4
Therefore, there are 4 ways to arrange 3 molecules on a symmetric polymer with 8 identical sites.
To determine the number of ways to arrange 3 molecules on a symmetric polymer with 8 identical sites, we can use the concept of permutations.
Permutations represent the number of arrangements of objects when the order matters and repetitions are not allowed. In this case, the order of the molecules on the polymer matters, but we assume that the molecules are identical, so repetitions are not allowed.
To find the number of ways to arrange the molecules, we can use the formula for permutations of n objects taken r at a time:
P(n, r) = n! / (n - r)!
Where "!" denotes the factorial function.
In our case, we have 8 identical sites on the polymer and we want to arrange 3 molecules. So we substitute n = 8 and r = 3 into the formula:
P(8, 3) = 8! / (8 - 3)!
= 8! / 5!
Calculating the factorials:
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
5! = 5 x 4 x 3 x 2 x 1 = 120
Plugging in the values:
P(8, 3) = 40,320 / 120
= 336
Therefore, there are 336 ways to arrange 3 molecules on a symmetric polymer with 8 identical sites.