I am trying to understand how a probability function is derived to Taylor series and how the factorials were factored to the final answer.
$$\frac{26!}{\sum_{k=1}^{26} \binom{26}{k}k!}\\ = \frac{26!}{ \sum_{k=1}^{26} \frac{26!}{k!(26-k)!}k!}\\ = \frac{26!}{\frac{26!}{25!}+\frac{26!}{24!}+\cdots +\frac{26!}{1!}}$$
I can understand the content of the above functions, however, i can't understand how it derives to:
$$= \frac{1}{\frac{1}{25!}+\frac{1}{24!}+\cdots+\frac{1}{1!}+1}$$
Why are all the numerators (26!) of the denominators changed to 1 (examples: $\frac{26!}{25!}$ changed to $\frac{1}{25!}$)?
kindly advise. Regards