# Normalized multinomial distribution

At some point, in Bishop's book 'Pattern recognition and Machine Learning', (p.75) he is talking about multinomial distributions in a classification context, introducing a suitable probability distribution $$p(\bf x | \mu)$$:

with given constraints for $$\bf x$$ and $$\bf \mu$$.

What I don't understand is why the distribution is normalized, i.e. equality 2.27. How does he achieve that?

Note that the possible values of $$\ \mathbf{x}\$$ are $$\ \mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_K\$$, where $$\ \mathbf{e}_j\$$ is the unit vector of $$\ \mathbb{R}^K\$$ whose $$\ j^\text{th}\$$ entry is $$1$$ and all of whose other entries are $$0$$. Thus, \begin{align} \sum_{\mathbf{x}}\prod_{k=1}^K\mu_k^{x_k}&=\sum_{j=1}^K\prod_{k=1}^K\mu_k^{\left(\mathbf{e}_j\right)_k}\\ &= \sum_{j=1}^K \prod_{k=1}^K\mu_k^{\delta_{jk}}\\ &= \sum_{j=1}^K\mu_j\\ &=1\ . \end{align}