In how many different ways $10$ identical ball can be distributed among $3$ children, if each receives atleast $2$ balls and no more than $4$ balls ?
Solution Distribute $10$ identical ball $\rightarrow 3$ distinct children
$$x_1+x_2+x_3 = 10$$ where $2\le x_i \le 4$
now I am not able to find how to approach by generating functions , since I am new to it please explain in detail
There is no need to use generating functions, the star and bars method suffices to solve the problem.
According to the question each child must get at least $2$ balls. Hence the answer to the question would be the number of non negative integral solutions to the equation $$ a+b+c=4$$ And coincidentally in none of the cases does a child get more than 4 balls, using the star and bars intuition. Hence the answer would be $$\binom {6}{2}=15$$
Method 2 :-
Each child must get either $2,3$ or $4$ balls and there are total four balls to be distributed among 3 children Hence the generating function would be -
We need to find the coefficient of $x^{10}$ in the expansion $$(x^2+x^3+x^4)(x^2+x^3+x^4)(x^2+x^3+x^4)=(x^2+x^3+x^4)^3= x^6 .(1-x^3)^3. (1-x)^{-3}$$
I hope you can take it from here.