PRML Bishop - Doubt in Entropy Formula (Section 1.6, equation 1.94)
You are not missing anything. You simply found another derivation of the same relationship.

We know that

$$^{n}C_{k} = \frac{n!}{k!(n-k)!} $$

If we expand the terms in your product, 

$$^{N}C_{1} * ^{N-1}C_{2} * ^{N-3}C_{3}* ^{N-6}C_{4} \cdot \cdot $$

, we get

$$\frac{n!}{1!(n-1)!} \cdot \frac{(n-1)!}{2!(n-3)!} \cdot \frac{(n-3)!}{3!(n-6)!} \cdot \frac{(n-6)!}{4!(n-10)!} \dots $$

If we simplify the factorials,

$$\frac{n}{1!} \cdot \frac{(n-1)(n-2)}{2!} \cdot \frac{(n-3)(n-4)(n-5)}{3!} \cdot \frac{(n-6)(n-7)(n-8)(n-9)}{4!} \dots $$

And now it's obvious that this is

$$ \frac{n!}{\prod_i i!} $$