Let $n \in \mathbb{N}, n \geq 4$ and $P = \prod_{r=0}^{n} \, ^nC_r$, then:
Step-by-Step Solution
Key Concept: Use the sum and product of roots along with the constraint that complex roots appear in conjugate pairs to systematically determine integer solutions.
For $f(x) = x^4 - 6x^3 + 26x^2 - 46x + 65$, complex roots occur in conjugate pairs: if $a_1 + ib_1$ is a root, so is $a_1 - ib_1$. From product of roots $= 65$ and sum $= 6$, with $a_1 + a_3 = 3$. Testing $a_1^2 + b_1^2 = 5$ and $a_3^2 + b_3^2 = 13$ gives $a_1 = 1, b_1 = 2$ and $a_3 = 2, b_3 = 3$. The roots are $1 \pm 2i, 2 \pm 3i$.
Correct Answer: I need to analyze the question about the product of binomial coefficients.
Given: $P = \prod_{r=0}^{n} \binom{n}{r}$ where $n \in \mathbb{N}, n \geq 4$
**Key insight:** The product of all binomial coefficients in a row has a known formula.