https://gateoverflow.in/65803/find-number-integral-solutions-using-generating-function
This post and material in its links are enough.
@bhuv very good analysis. But small correction is required in your explanation (check for typo)
$\dfrac{1}{(1-x)^4}:\dfrac{1}{(1-x)^n}=\sum_{k=0}^{\infty }\ ^{n-1+k}C_{k}\ x^k$
$(1-x^6)^4:(1-x^r)^n=\sum_{k=0}^{n}\ ^{n}C_{k}\ (-x^r)^k$