In Bishop's PRML Book it describes multinomials as:
"If we denote probability $x_k = 1$ by probability $u_k$ then the distribution of $x$ is given by: $p(x|\mu) = \Pi_{k=1}^{K}\mu_{k}^{x_k}$ where $\pmb{\mu} = (\mu_{1}...\mu_{k})^T$" where earlier they defined that say if a variable can take on $K$ states, then in a case where $k=6$ and a variable corresponds to the state where $x_3=3$, then let $x = (0, 0, 1, 0, 0, 0)^T$
I think I understand the concept of how a multinomial works but the notation in this book really confuses me. I'm not sure why there is a product symbol in $p(x|\mu)$ - isn't the probability of some state x happening just it's probability as given by $\mu$. Could anyone explain in more detail what all this means?
EDIT: Here I included a screenshot of the passage.
What Bishop is describing is a $K$-category analog of a Bernoulli random variable. Just as Bernoulli is to the binomial distribution, so is this thing that Bishop describes here is to the multinomial distribution.
The product that puzzles you is the natural generalization of what happens in the Bernoulli case. Say you flip a coin $X$, just once, where "tails" is coded with a $0$ and "heads" with a $1$, and where $P(X=1)=p$ and $P(X=0)=q$, with $p+q=1$. These two probability equations can be packed into one as follows: $P(X=x) = p^x q^{1-x}$. If $x=0$ the product $p^xq^{1-x}$ works out to $p^0q^1=q$, and if $x=1$ the product $p^xq^{1-x}$ works out to $p^1q^0=p$.
Now Bishop goes on to say, look, lets code the result of the coin flip with a vector $\mathbf x = (x_0,x_1) = (1-x,x)$ which is equal to $(0,1)$ if $x=1$ and is equal to $(1,0)$ if $x=0$. Here the idea is that $x_0$ counts the number of times (which is either $0$ or $1$) in the single flip that tails came up, and $x_1$ counts the number of times (also $0$ or $1$) in the single flip that heads came up. So the new version of the product formula for the probability of outcome $\mathbf x$ is $p^{x_1}q^{x_0}$. And so on.
An awful lot of notation for an almost trivial idea: I don't blame you for getting confused.