<s>[INST] Calculate how many ways to get change of 78 I been asked to calculate how many ways there are to get change of 78 cents with the coins of 25,10,5,1.
I been able to write this down:
$25a + 10b + 5c + d = 78$
But I do not know how to continue. Can you help me please?
 [/INST] Consider the product in alex.jordan's answer. The product of the last two factors is
$$1 + x^{10} + x^{20} + x^{25} + x^{30} + x^{35} + x^{40} + x^{45} + 2 x^{50} + x^{55}
+ 2 x^{60} +  x^{65} + 2 x^{70} + 2 x^{75} + \ldots)$$
The product of the first two is
$$ 1 + \ldots + x^4 + 2 (x^5 + \ldots + x^9) + 3 (x^{10}+\ldots+x^{14}) + \ldots
+ 16 (x^{75}+\ldots+x^{79})+\ldots$$
So the coefficient of $x^{78}$ is 
$$ \eqalign{ 16 & \text{ for } 1 \times 16 x^{78}\cr
+ 14 & \text{ for } x^{10} \times 14 x^{68}\cr
+ 12 & \text{ for } x^{20} \times 12 x^{58}\cr
+ \ldots\cr
+ 2 & \text{ for } 2x^{75} \times x^3}$$ 
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