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lang-mma
Simplifyis irrelevant. This sort of problem has been noted before. Example: mathematica.stackexchange.com/questions/78945/… $\endgroup$BinomailandFactorial. With the correctionSum[ (m - 2)!/((k - 2)! (m - k - 1)!), {k, 2, m - 2, 2}, Assumptions -> m/2 \[Element] Integers && m > 10]also returns 0 which is wrong. It appears that there is indeed a bug here. I have removed the original answer. $\endgroup$