# How to deduct the coefficient of the Dirichlet distribution?

I am studying the textbook Introduction to Probability Models by Sheldon Ross. In Section 3.6 page 151 (10th edition), the author uses the Dirichlet distribution to deduct the coefficient of a "uniform" distribution. The "uniform" distribution is

$$f(p_1,p_2,\cdots,p_m) = \left\{\begin{array}{rl} c, & 0\leq p_i \leq 1, i = 1,\cdots, m, \sum_1^mp_i=1 \\ 0, & \text{otherwise} \end{array} \right.$$

So I check the wikipedia page about Dirichlet distribution. Among all the material I find out that "uniform" distribution (with constraint on the sum of variables) follows a symmetric Dirichlet distribution with $\alpha$'s become identical. $$f(x_1,\dots, x_{K-1}; \alpha) = \frac{\Gamma(\alpha K)}{\Gamma(\alpha)^K} \prod_{i=1}^K x_i^{\alpha - 1}$$ Thus it gives me the result as the author indicate $c=(m-1)!$.

The question is, to me, how to get the coefficient without referring to the Dirichlet distribution or get the coefficient given the kernel of the Dirichlet distribution, of course with the constraint that $\sum x_i = 1$.

I find out a post in the same site, which is to Construct the Dirichlet distribution using Gamma RV's. Part of the solution can answer this question. However, it maybe of some value to help others search on this topic, so I decide not to remove the post.

Since that version is a little bit hard to understand, Can you give a more plain explanation?