I feel the above formula needs a little more explanation. Say we have a random variable X, where X~Bin(N,p). This means we are counting the number of successes in N trials, where each trial has a probability p of success. We also define q as 1-p, in other words the probability of failure. We are interested in the probability that X takes the particular value x. To calculate this, we use:
P(X = x) = NCn x pn x qN-n
You see we have three terms here. First, the term pn is the probability of n success. Simple enough. Now if we have n successes, we must have N - n failures to make up the N trials. So we have the term qN-n for the probability of N-n failures. We multiply these two probabilities together because we need both of the events to occur.
Now the term NCn. Perhaps you recognise the C as the Choose function, but perhaps not. Put simply, aCb is the number of ways of choosing b objects from a, where the order in which you pick doesn't matter. Remember, when we have n successes and N - n failures, they can happen in any order; we don't mind. Therefore the Choose term effectively represents the number of different orders in which the successes and failures can occur.
Note that we could equally have chosen the number of failures, and put NCN - n at the front. However, this is exactly the same! By choosing N - n failures, you automatically choose n successes. You can see this in symmetry of Pascal's Triangle, but that's another story. Conventionally, we use the one that is shorter to write. Mathematicians like brevity.