My professor gave us this problem. May you give me your thoughts about my answer and the problem?
Thank you so much!
Here is the problem:
If object A can be chosen with repetition either not at all or twice, object B either once or three times, and object C not at all or once, how many selections of thirteen objects can be made?
I want to explain my reasoning about the generating functions that I selected.
Generating Function for object A:
As object A can be selected with repetition either not at all or twice,
without repetition would be $(1+x^2)$ ($1$ for zero A's, $x^2$ for two A's).
I am assuming that the second time that we choose object A, we can have zero or $4$ A's, then we have $(1+x^2+x^4)$, the third time we have $(1+x^2+x^4+x^6)$ and the maximum quantity of A's would be 12 because, we will have at least one object B.
Thus the generation function for object A would be:
Generating function for object B:
As object B can be selected either once or three times, I will be using x for once and $x^3$ for three times. Then the generating function for object B is:
$(x + x^3)$
Generating function for object C
As object C can be selected not at all or once, I am using
Finally, I multiply all of them, using Wolfram Alpha:
My answer is the coefficient of x^13 that is 2.
I want to try a
The possible values for A are 0,2,4,6,8,10 and 12.
The possible values for B are 1, 3
The possible values for C are 0, 1
The combinations for B and C are only four:
0 C's + 1 B's (Subtotal 1 object, the only possible values for A is 12)
0 C's + 3 B's (Subtotal 3 objects, the only possible value for A is 10)
1 C's + 1 B's (Subtotal 2 objects, there are no possible values for A to complete 13)
1C's + 3 B's (Subtotal 4 objects, there are no possible values for A to complete 13)
This approach also confirms that there are only 2 options.