Integration (1)

$\begin{aligned} \frac{\displaystyle \int_0^1 (1-t^{50})^{100} dx}{\displaystyle \int_0^1 (1-t^{50})^{101} dx} & = \frac{ \frac{1}{50} \displaystyle \int_0^1 x^{-\frac{49}{50}}(1-x)^{100} dx}{ \frac{1}{50} \displaystyle \int_0^1 x^{-\frac{49}{50}}(1-x)^{101} dx} \quad \quad \small \color{#3D99F6}{\text{Let }x = t^{50} \Rightarrow dx = 50 t^{49}dt \Rightarrow dt = \frac{x^{-\frac{49}{50}}dx}{50}} \\ & = \frac{\displaystyle \int_0^1 x^{\frac{1}{50}-1}(1-x)^{101-1} dx}{ \displaystyle \int_0^1 x^{\frac{1}{50}-1}(1-x)^{102-1} dx} \\ & = \frac{B\left(\frac{1}{50},101\right)}{B\left(\frac{1}{50},102\right)} \quad \quad \small \color{#3D99F6}{B(x,y) \text{ is Beta function}} \\ & = \frac{\Gamma\left(\frac{1}{50}\right)\Gamma\left(101\right)}{\Gamma\left(101\frac{1}{50}\right)} \times \frac{\Gamma\left(102\frac{1}{50}\right)}{\Gamma\left(\frac{1}{50}\right)\Gamma\left(102\right)} \quad \quad \small \color{#3D99F6}{\Gamma(x) \text{ is Gamma function}} \\ & = \frac{101\frac{1}{50}}{101} \\ & = \frac{5051}{5050} \end{aligned}$

$\Rightarrow p-q = 5051-5050 = \boxed{1}$