Calculating n!/(k-1)!(n-k+1)! from Binomial Theorem

[Jef123]

1. How do you get n!/(k-1)!(n-k+1)! from $$\begin{pmatrix} n\\k-1 \end{pmatrix}$$

I thought it would be n!/(k-1)!(n-k-1)! where the n-k+1 on the bottom of the fraction would be a n-k-1 instead. I don't understand why there is a "+1" wouldn't you just replace k with k-1 in the binomial formula?

 

[whoareyou]

$$\binom{n}{k-1}=\frac {n!}{(k-1)!(n-(k-1))!}=\frac{n!}{(k-1)!(n-k+1))!}$$

 

[Jef123]

I just got it after i typed that out. I can't believe i didnt notice that

 

[Alicelewis11]

Binomial theorem use for different purposes. The above person share write formula.