I'm doing a bit of exam revision and came across the following question. I would like to know if my working (thus answer) is correct, or where I've gone wrong.
Four stores are getting new lawnmowers in. Each outlet must get at least 5 lawn-mowers. The two small outlets can cope with at most 20 lawnmowers, while the larger two can cope with at most 29. A shipment containing 60 lawn mowers has arrived. How many ways can they be distributed?
My working goes as follows:
Give each store 5 of the mowers to satisfy the first condition. There are now 40 mowers to distribute, resulting in the generating function
$f(x) = \left(1 + x + x^2 + \cdots + x^{15}\right)^2\left(1 + x + \cdots + x^{24}\right)^2$
Then solve for the coefficient of $x^{40}$
$f(x) = \left(1 - x^{16}\right)^2\left(1 - x^{25}\right)^2\left(1 - x\right)^{-4}$
Expanding this function gives:
$ \left({2 \choose 0} - {2 \choose 1}x^{16} + {2 \choose 2}x^{32}\right)\left({2 \choose 0} - {2 \choose 1}x^{25} + {2 \choose 2}x^{50}\right)\left(\sum\limits_{i=0}^n {-4 \choose i}x^i\right)$
Then multiplying the factors to get $x^{40}$ terms gives:
${43 \choose 40} - {2 \choose 1}{27 \choose 24} -{2 \choose 1}{18 \choose 15} + {2 \choose 2}{11 \choose 8} = 5024 $
Is this correct, if not could you let me know where I went wrong.
Thanks