Equal coefficients

Question

$\begin{align*} &A = \left ( p x + q \right )^{504} \text{ where p and q are +ve integers and }gcd(p,q) = 1 \\ &\text{Given } \left [ x^4 \right ] = \left [ x^5 \right ] , \text{ where } \left [ x^k \right ] \text{ denote the coefficient of }x^k \text{ in A.} \\ &\text{What is p+q ?} \end{align*}$

Comments

101??

---

As, $\left [ x^{4}\right ]=\left [ x^{5} \right ]$

$\binom{504}{4}.p^{4}.q^{500}=\binom{504}{5}.p^{5}.q^{499}$

=$\binom{504}{4}.q=\binom{504}{5}.p$

$\frac{p}{q}=\frac{1}{100}$

So, p+q=101