Generating Functions and stars and bars

Question

In the expansion of

$(1+x+x^{2}+------x^{27})(1+x+x^{2}------x^{14})^{2} $

what is the coefficient of $x^{28}$?

Can someone please tell me what I did wrong

This can be interpreted as a 28 sided die with numbers from [0,27] and 2 15 sided dice's with numbers from [0,15] and we are trying to find the number of ways these dices can add up to 28

So now using stars and bars we have the following

$27-a+14-b+14-c=28$

Now that equals

$a+b+c=27$

So now the answer is ${28 \choose 2}=378$

Answer 1

The ones you overcounted were the solutions to

$$ a+b+c=27\\ b>14 \; \text{or} \; c>14 $$

Counting the number of solutions where $b>14$ gives

$$ a + (b-14) + (c) = 27 -14 $$

Counting the number of solutions where $c>14$ gives

$$ a + b + (c-14) = 27 -14 $$

There are no solutions with both $b,c>14$.

So do stars and bars on these two cases and subtract it off from your previous answer.