)The purpose of summing up the series $\sum^{n}_{x=2}x(x-1)\binom{n}{x}p^{x}q^{n-x}$ is to determine the total number of successful outcomes in a binomial experiment with n number of trials, where x number of successes occur and p is the probability of success.
This series is related to binomial theorem because it involves the combination of two binomial coefficients, $\binom{n}{x}$ and $\binom{n}{x-1}$, and the terms within the parentheses can be simplified to (x-1) and x, which are also coefficients in the binomial expansion.
This series can be used in real-life situations to calculate the probability of a certain number of successes in a series of trials, such as the probability of getting 3 heads in 5 coin tosses or the probability of rolling a total of 8 on two dice in 10 rolls.
The parameters p and q represent the probability of success and failure, respectively, in a binomial experiment. They are used to calculate the probability of a certain number of successes, x, in n trials. The value of q is equal to 1-p, as there are only two possible outcomes in a binomial experiment.
Yes, there is a formula to calculate the sum of this series, which is $\sum^{n}_{x=2}x(x-1)\binom{n}{x}p^{x}q^{n-x} = np^2 + n(n-1)p^2q + n(n-1)(n-2)p^2q^2$. This formula can be derived using the binomial theorem and simplifying the terms within the parentheses.