Combinations for Negative n

If  
\[m\]
  is negative then  
\[{}^nC_k\]
  may still have a value. If fact  
\[{}^nC_k\]
  has a value if  
\[| n | \gt 0\]
, and
\[{}^{- n}C_k=(-1)^k{}^{n+k-1}C_k\]
  for  
\[n \gt 0\]
.
Example:  
\[{}^{-5}C_3 = (-1)^3 {}^{5+3-1}C_3=-{}^7C_3 = -\frac{7!}{3!4!}=-35\]

Example:  
\[{}^{-3}C_5 = (-1)^5 {}^{3+5-1}C_5=-{}^7C_5 = -\frac{7!}{5!2!}=-21\]

Add comment

Security code
Refresh