In the book "Concrete Mathematics" by Knuth there is a binomial coefficient identity
$\sum_{k} {a+b \choose a+k}{b+c \choose b+k}{c+a \choose c+k}(-1)^k=\frac{(a+b+c)!}{a!b!c!}$
but no proof is given. Can someone at least guide me to a way to prove it?
This is Dixon's summation theorem for terminating ${}_{3}F_{2}$ hypergeometric series. A very elementary self-contained proof is given in my article "An algebraic independence result related to a conjecture of Dixmier on binary form invariants". The relevant part is from the bottom of page 5 to the middle of page 8 and no need to read anything else in the article.