[Math] Binomial coefficient equal to $\binom nk + 3\binom{n}{k-1} + 3\binom{n}{k-2} + \binom{n}{k-3}$

Question

Find one binomial coefficient equal to the following expression:

$$\binom nk + 3\binom{n}{k-1} + 3\binom{n}{k-2} + \binom{n}{k-3}$$

I tried to expand using the definition of $\dbinom{n}{k} = \dfrac{n!}{k!(n-k)!}$, but it was unwieldy. Which identities should I use?

Answer 1

You're looking at: $$\binom 30\binom nk + \binom31\binom{n}{k-1} + \binom32\binom{n}{k-2} + \binom33\binom{n}{k-3}$$ Apply Vandermonde's identity.