If the value of the definite integral $\int_0^1 {^{207}}C_x x^{200}(1-x)^7 dx$ is equal to $1/k$ where $k \in \mathbb{N}$, then the value of $k/26$ is ____.
Step-by-Step Solution
Key Concept: Use repeated integration by parts to reduce the power of $(1-x)$ while increasing the power of $x$ until the boundary term vanishes.
The integral $I = ∫_0^1 C_7^{207}x^{200}(1-x)^7dx$ is evaluated using repeated integration by parts. After the first integration by parts, the $(1-x)^7$ term becomes zero at the boundary, leaving $\frac{7}{201!}∫_0^1(1-x)^6·x^{201}dx$. Repeating this process 6 more times yields $I = C_7^{207}·\frac{7!}{201·202·203·204·205·206·207}·\frac{1}{208} = \frac{(207)!·7!}{(207)!·7!·208·k}$, which simplifies to $k = 208$.
Correct Answer: 8