Binomial + Binomial


I'm having difficulty working with the binomial - specifically aggregating binomial distributions.

Hoping somebody can tell me where I'm going wrong.

Let me give a totally fictional example:
In a soccer match, there are 40 attempts at scoring a goal. Of these, 5 go in. Hence the trials = 40 and p(success) = 12.5%

This is made up by both team A and team B having 20 attempts each, team A score 3 (15% success), team B score 2 (10% success).

If this scenario were to repeat, with the same expectations for each team and totals. Using the total figures, the probability of there being 3 goals in total is 13.797% (binomial with trials 40, success 12.5%).

The only way to achieve 3 goals in total is the following combinations between team A and team B:
Using the binomial based on the teams expectations (20 trials and 15% or 10% success respectively) to calculate the probability for each scenario, the sum of these equates to 13.786%

Shouldn't these equate to the same figure? Would really appreciate someone telling me where I'm misunderstanding my stats :)

P.S. If this sounds an interesting world of real-world stats to you, it would mean the world to me if you would have a look at my query from a month ago that has yet to be attempted by anyone. After 2 months trying, I'm no closer to solving the problems: