<s>[INST] Finding the closed formula of a generating function for flower selection Find the generating function for the number of ways to select $n$ flowers from a boundless supply of dandelion, tulip, rose and blossoms if we have at least one flower of each type, the number of dandelions is less than 4, the number of tulips is a multiple of 5, and the number of roses are a multiple of 2? Find generating function in CLOSED form.
So we have:
$$
F_1(x) = x + x^2 + \dots = \frac{x}{1-x}
$$
$$
F_2(x) = x + x^2 + x^3 = \frac{x(1-x^3)}{1-x}
$$
$$
F_3(x) = x^5 + x^{10} + \dots = \frac{x^5}{1-x^5}
$$
$$
F_4(x) = x^2 + x^4 + \dots = \frac{x^2}{1-x^2}
$$
I'm not sure what it means to take it closed form from here? Do we just multiply them together and we are done?
 [/INST] Yes, the desired generating function is the product
$$\frac{x^9 (1-x^3)}{(1 - x)^2(1-x^2) (1-x^5)}=\frac{x^9 (1+x+x^2)}{(1+x) (1 - x)^3 (1+x+x^2+x^3+x^4)}.$$
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