Likelihood of Two Binomial Distributed RV's

Question

We are given that Let X1~Bin(n1 = 34, p1) and X2~Bin(n2 = 56, p2)

In general, what is the likelihood, L(p1, p2) = f (X1, X2 | p1, p2) for the data X1 and X2

I believe that I am supposed to use a binomial here, adding together the two RV's. I understand that If X1~Bin(n,p) and X2~Bin(m,p), then X1+X2~Bin(n+m,p). Here however, I am given that X1 has prob=p1 and X2 has prob=p2. When adding them together, what is the new probability?

Answer 1

$$L(p_1,p_2)={n_1\choose x_1}p_1^{x_1}(1-p_1)^{n_1-x_1}{n_2\choose x_2}p_2^{x_2}(1-p_2)^{n_2-x_2}$$