Binomial Distribution with one trial having different probability

How do I (correctly and easily) factor in one trial that has a different probability? For example:

I flip a coin 5 times (50/50). But, on the first flip, I use my special weighted coin that hits HEADS more often, say (80/20).

What are the Odds of each result, # heads: (0,1,2,3,4,5)

Thanks everyone.

If we say that the first coin flip is event $A$, and the second is $B$, and with $X$ as the overall outcome as a combination of $A$ and $B$, then $$\mathbb{P}(X=k) = \mathbb{P}(A=0) \times \mathbb{P}(B=k) + \mathbb{P}(A=1) \times \mathbb{P}(B=k-1)$$