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?

  • 5
    $\begingroup$ I suppose $$ \binom{\binom nk + 3\binom{n}{k-1} + 3\binom{n}{k-2} + \binom{n}{k-3}}{1} $$ does not count. =) $\endgroup$ – Srivatsan Oct 30 '11 at 2:38

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.


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