Suppose X and Y are independent Poisson random variables with parameters a and b respectively. Find the joint mass function

P[X = k, Y = m]

I got

(exp(-a).a^k)/k! x (exp(-b).b^m)/m!

Find P[X + Y = n]

I got = P[X = k, Y = n - k]

= sum over k to n (exp(-a).a^k)/k! x (exp(-b).b^n - k)/(n-k)!

But I'm not sure how to complete

is this right so far?

thanks