Deriving the binomial distribution formula

[Saladsamurai]

I am trying to follow along with this derivation of the binomial distribution formula:
b(x;n,p) = nCx*p^xq^n-x

But I do not really understand the meaning of the part on bold. What is this "specified order" business now? I feel like I am missing something big here.

Let us now generalize the above illustration to yield a formula for b(x;n,p). That is, we wish to find a formula that gives the probability of x successes in n trials for a binomial experiment. First, consider the probability of x successes and n — x failures in a specified order. Since the trials are independent, we can multiply all the probabilities corresponding to the different outcomes. Each success occurs with probability p and each failure with probability q = 1 — p. Therefore, the probability for the specified order is p^xq^n-x. number of sample points in the experiment that have x successes and n — x failures. This number is equal to the number of partitions of n outcomes into two groups with x in one group and n—x in the other and is written nCx as introduced in Section 2.3. Because these partitions are mutually exclusive, we add the probabilities of all the different partitions to obtain the general formula, or simply multiply p^xq^n-x by nCx.

 

[Daft]

Think about what the "nCx" part of the formula does.

Let's say we're rolling a regular dice 4 times, and we want the probability of a 6 only once. We know this would consist of 1 success and 3 failures ( \frac{1}{6}\frac{}{} )( \frac{5}{6} )³, however it does not account for which roll we get the 6, since it can occur in any 1 out of the 4 rolls, we multiply by 4C1.

So that "specific order" business is to eliminate the need to account for exactly when our successes occur

Does that make sense?