I am asked
How many ways are there to distribute 10 identical balls among 2 boys and 2 girls, if each boy should get at least 1 ball and each girl should get at least 2 balls? Express the answer as a coefficient of a suitable power of x in a suitable product of polynomials.
I am unable to understand how I can use products of polynomials to answer this question and how to proceed.
Ive set up something like
$(B^1+B^2+B^3+B^4+B^5)^2(G^2+G^3+G^4+G^5)^2$
But I do not know when to terminate each factor as to know when I have allocated all 10 balls-- I don't want to go over the 10 balls.
To work over generating functions, you must write the variables as the same. For example , use only $x$ instead of $B$ and $G$.When we comes to your problem such that "But I do not know when to terminate each factor as to know when I have allocated all 10 balls". For this question , the terminal point is not important , the generating function will handle it automatically.
For easier calculation , you can write them like $$\bigg(\frac{x}{1-x} \bigg)^2 \times \bigg(\frac{x^2}{1-x} \bigg)^2. $$
What you need to do is just finding the coefficent of $x^{10}.$
Whats more , as you see the foregoing expression can be writen by $$x^6 \times \bigg(\frac{1}{1-x}\bigg)^4,$$ so just find the coefficient of $x^4$ in $$\bigg(\frac{1}{1-x} \bigg)^4.$$
You can use wolfram-alpha for it!