Sum of independent random variables (conditional prob.)

Question

When $X_1,...,X_{k+l}$ are independent random variables with $P(X_n=1)=p$ and $P(X_n=0)=1-p$, ($n=1,..,k+l$), and $B=X_1+...+X_k$ and $C=X_1+...+X_{k+l}$, what is $P(C=n|B=m)$ for $n=0,...,k+l$ and $m=0,...,k$?

My ideas:

$A:=X_{k+1}+...+X_{k+l}$

$P(C=n|B=m)= P(A+B=n|B=m)=P(A=n-m|B=m)$

I think A and B are independent (?), so i can write $P(A=n-m|B=m)=P(A=n-m)$, but what now (if this is right)?

Answer 1

1. $A$ and $B$ are independent because they are sums of different independent variables.

2. $A$, $B$ and $C$ are binomial RVs