How can I simplify this expression involving summation and factorials?

[Polymath89]

I need to simplify this expression and I don't know how to deal with the factorials in the sum

$$e^{-(\lambda + \mu)}\sum_{k=0}^w \frac{\lambda^k \mu^{(w-k)}}{k!(w-k)!}$$

Can anybody give me a hint on how to sum over the factorials?

 

[Mark44]

I need to simplify this expression and I don't know how to deal with the factorials in the sum

$$e^{-(\lambda + \mu)}\sum_{k=0}^w \frac{\lambda^k \mu^{(w-k)}}{k!(w-k)!}$$

Can anybody give me a hint on how to sum over the factorials?

This is already pretty simplified. Do you mean that you want to expand it?

 

[Polymath89]

Sorry for not being clear, yes I want to expand the sum.

 

[mathman]

The sum looks almost like a binomial expansion.
net result e^-(λ+μ)(λ+μ)^w/w!.

 

[Polymath89]

The sum looks almost like a binomial expansion.
net result e^-(λ+μ)(λ+μ)^w/w!.

Thank you very much.