For distributing Apples:
$\huge [x^{77}] (x + x^2 + x^3 + ...)^{44}$
How I get this?
Each child will get atleast 1 apple, so series start from $x$ and not 1(because 1 means $x^0$, i.e. a child getting 0 apples). Now we have to do this for 44 children, so this series multiplied 44 times. And now in this we have to find coefficient of $x^{77}$.
Solving it:
$ = [x^{77}] (x + x^2 + x^3 + ...)^{44}$
$ = [x^{77}] x^{44}(1 + x + x^2 + ...)^{44}$
$ = [x^{33}] (x + x^2 + x^3 + ...)^{44}$
$ = [x^{33}] (\frac{1}{1-x})^{44}$
$ = [x^{33}] (1-x)^{-44}$
$ = [x^{33}] (\sum \binom{-44}{r} (-x)^r)$
$ = \binom{-44}{33}(-1)^{33}$
$ = \binom{76}{33}$
This you can verify. (As we do without generating functions.)
Now, for distributing oranges:
$\huge [x^{66}] (1 + x + x^2 + ...)^{44}$
Here there is no limitation on min number of oranges every child can get, so series starts from $x^0$.
Solving it:
$ = [x^{66}] (1 + x + x^2 + ...)^{44}$
$ = [x^{66}] (\frac{1}{1-x})^{44}$
$ = [x^{66}] (1-x)^{-44}$
$ = [x^{66}] (\sum \binom{-44}{r} (-x)^r)$
$ = \binom{-44}{66}(-1)^{66}$
$ = \binom{109}{66}$
Total ways = Ways to distribute apples * ways to distribute oranges
$ = \binom{76}{33} * \binom{109}{66}$
PLEASE TELL ME IF THERE IS ANY MISTAKE.
Ask if any doubt.