No. of integral solutions of an equation with upper and lower bound without 'generating the function' method

Question

Question from my book

Now, I know how to solve this using 'generating the function' method. But I cant figure out what method is used in the solution of the problem given in the book,especially from the part where variables have been changed. Can you please explain to me the method used here? Solution

Answer 1

Well, here is a simpler solution w/o too many variable changes. You can then study the solution and grasp that, too

$12$ identical apples are to be distributed between $4$ children with each getting a minimum of $1$ and a maximum of $4$

Give one apple to each child, so now we have only $8$ apples to distribute, which could have been done in $\binom{8+4-1}{4-1} = \binom{11}3$ ways without the upper limit.

To take care of the upper limit, applying inclusion-exclusion,

$\binom{11}3$ - (at least one child now gets more than $3$ apples) + (at least two children now get more than $3$ apples)

$=\binom{11}3 - \binom41\binom73 +\binom42\binom33$

Answer 2

Let us consider the possible numbers of apples, sorted increasingly (there are few possibilities, we can do it explicitly).

If the smallest number is $1$, we need to assign eleven apples to three children, and the only way is $344$. By continuing the reasoning, we quickly find the only options

$$1344, 2244, 2334, 3333.$$

Now we can assign these numbers to specific children, by permutation. But when there are equal numbers, we divide by the numbers of permutations of these equal numbers. That gives us,

$$\frac{4!}{2!}+\frac{4!}{2!\,2!}+\frac{4!}{2!}+\frac{4!}{4!}=12+6+12+1=31.$$