1. The problem is understanding why the factorial expression $(N+1)!$ remains as is, instead of simplifying to $(N+1) - 1$. 2. Recall the definition of factorial: for any positive integer $k$, $$k! = k \times (k-1) \times (k-2) \times \cdots \times 2 \times 1.$$ 3. Specifically, $(N+1)!$ means: $$ (N+1)! = (N+1) \times N \times (N-1) \times \cdots \times 2 \times 1.$$ 4. It is important to note that factorial is a product, not a subtraction or addition. So you cannot simplify $(N+1)!$ as $(N+1) - 1$. 5. If you "unravel" $(N+1)!$, you write it as: $$ (N+1)! = (N+1) \times N! $$ which shows the factorial expanded one step. 6. Therefore, $(N+1)!$ stays as a factorial expression or is expanded as above, but never simplified to $(N+1) - 1$. 7. In summary, factorial means multiplication of all positive integers up to that number, not subtraction.