If $\sum_{k=0}^{n}\left(\binom{n}{k}\cdot\binom{m}{Ck-Zn}\right) = K\binom{m}{Cn}$, then $K$ is equal to
Step-by-Step Solution
Key Concept: The binomial theorem states $(x+y)^n = \sum_{r=0}^{n} {}^nC_r x^r y^{n-r}$
We use the binomial theorem expansion. Starting with LHS $= \sum_{r=0}^{n} {}^nC_r \cdot 2^r \cdot {}^nC_n$. This simplifies using combinatorial identities and properties of binomial coefficients. After expanding and simplifying, we get $K = 2^n$.
Correct Answer: 1