The sum of a linear combination of independent normal variables is also normally distributed, with mean equal to the sum of the linear combination of the means and variance the sum of the variances times the linear coefficients squared. So the distribution of planes getting through, given your comments on my original answer, must be have a normal distribution with mean:
If there is any dependence between thre three SAM batteries' performance - eg on particular weather conditions, all batteries have a bad day - you can take this into account too extending to three variables the standard results for sums of dependent random variables:
$Var(X_1+X_2)=Var(X_1) + Var(X_2) +2Cov(X_1X_2)$
My scepticism about this as a model remains - certainly if your numerical example is at all representative of the sort of numbers involved - so I leave my original answer below. If your numerical example is representative, the number of planes getting through would be normally distributed with a mean of around 2 or 3 and a variance of about 150 - 40% of the distribution in the (impossible) negative zone, and another 25% indicating more planes get through than the ten that started! So this might be why it seemed to you that it should be more complicated than just the sum of normally distributed variables.
I think the problem is misconceived because of the paragraph:
Assume each SAM battery shoots down a random number of planes. This
random number is drawn from a normal distribution unique to each
battery i.e. Mu_A, Var_A, Mu_B, Var_B etc.
It is not possible for the number of planes shot down to be exactly normally distributed, because the number shot down has to be an integer, and a normal distribution is continuous. Further, unless the number of planes going through is very large and the number shot down relatively small, it seems unlikely that the number shot down will be even approximately normally distributed. Finally and most importantly, surely the number shot down is not going to be characterised just by SAM-specific parameters (Mu_A, Var_A) etc but also by the number of planes going through.
A more plausible (but still very simplifying) model would be that each SAM battery shoots down a random number of planes from a binomial distribution, characterised by $p_A$, $p_B$, etc (SAM-specific parameters) and $N$ - the number of planes that reach that battery. I say this is still inadequate because there would be diseconomies of scale - if the SAM is swamped with 1000 planes at once presumably the proportion it hits will be less than if it only has 10 or so to focus on. But for your problem (judging from your numerical example), I would say this is a much better way of looking at than as normally distributed.
Reconceived this way, the chance any single plane has of getting through all three SAM batteries (assuming - again implausibly because of pilot and plane skill etc, but maybe ok for now - that these are three independent events) is
You could use this fact to derive the actual distribution you are interested in of the number of planes that go through (I won't do this for you as it looks like homework).
Now, if I've missed the point and the "normal distribution" is an essential part of the problem, everything I've written above is unhelpful. But the normal distribution approach could only work if the number of planes shot down is not particularly dependent on the number there are to shoot.