Find the value of $\dfrac{1}{\left(n-1\right)!}+\dfrac{1}{\left(n-3\right)!3!}+\dfrac{1}{\left(n-5\right)!5!}+...$

∵1!=1

∴ The given series can be written as

$\dfrac{1}{\left(n-1\right)!}+\dfrac{1}{\left(n-3\right)!3!}+\dfrac{1}{\left(n-5\right)!5!}+...$+n

∵ sum of values of each terms in fraction are equal

i.e., (n−1)+1=(n−3)+3=(n−5)+5=........

From (1)

$\frac{1}{n!}\ [\frac{n!}{(n−1)!1!}+\frac{n!}{(n−3)!3!}+\ \frac{n!}{(n−5)!5!}+........]$

=$\frac{1}{n!}(nC1+nC3+nC5+......)$

=$\frac{2^{n-1}}{n!}$