Understanding The Binomial Probability Mass Function Formula

The probability mass function for a binomial random variable X is given by,

P(x) = nCx p^x * q^(n-x)

This is the probability of exactly x successes among n trials in a binomial experiment where,
p denotes the probability of success
q denotes the probability of failure
n denotes the number of trials
x denotes the number of success
nCx is the number of ways to choose x objects from a group of n without regard to order


Ok so how does this formula make sense?

So we have n spots,

_ _ _ _ ... _ _ _

We want to choose x of them for our successes, and there are nCx ways to do that.


Ok we want x successes, by the multiplication rule that means we have,

p*p*...*p <--- x copies of p
p^x.

That leaves n-x spots, those must be the failures,
q*q*...*q <--- n-x copies of q
q^(n-x)

Put it all together, and we have

nCx p^x * q^(n-x)

Simplistic but it should help provide some insight!