<s>[INST] Summation of factorial. $$2(\frac{1}{3!\times7!}+\frac{1}{1\times9!})+\frac{1}{5!\times5!}=\frac{2^a}{b!}$$ find $a,b$ by some predictions I see $b=10$ but what about numerator. I think we have to $\sum {N\choose r}=2^N$ but how?. Thanks.
 [/INST] Firstly write it as follows:
$$\frac{1}{1!\times9!}+\frac{1}{3!\times7!}+\frac{1}{5!\times5!}+\frac{1}{7!\times3!}+\frac{1}{9!\times1!}$$
$$=\sum_{n=0}^4\frac{1}{(2n+1)!(10-(2n+1))!}$$
$$=\frac{1}{2}\sum_{n=0}^{10}\frac{1}{n!(10-n)!}$$
$$=\frac{1}{2}\times\frac{2^{10}}{10!}$$
$$=\frac{2^9}{10!}$$
Hence $a=9$, $b=10$.
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