What is $(3n+3)!$ equal to

Is $(3n+3)!$ equal to:

a) $(3n+3)\cdot(3n)\cdot(3n-3)\cdot(3n-6)\cdot...\cdot(1)$

b) $(3n+3)\cdot(3n+2)\cdot(3n+1)\cdot(3n)\cdot...\dot(1)$

I was wondering, since $$(n+1)!=(n+1)\cdot n!$$ Shouldn't the right option be a)? Because,

$$(3n+3)!$$

using the definition of the factorial equal to

$$(3n+3)!=(3n+3)\cdot(3(n-1)+3)!=(3n+3)\cdot(3n)!$$

Solution 1:

The factorial function receives $Q$ and returns $Q(Q-1)(Q-2)\dots 2\cdot 1$. This is $(3n+3)(3n+3-1)(3n+3-2)\dots 2\cdot 1$ for $3n+3=Q$.

Note that variables such as $n,x,y$ etc. are used repeatedly in various contexts but without keeping the same values in between. The point of using $n$ over and over again is often to signal that it refers to an integer (but usually not the same integer from one context to another).