Properties of C(2n,n)

☀ C(2n,n) = (2n)! / (n!)2 = 2n *(2n-1)(2n-3)(2n-5)...........1 / n!  ( no. of terms in red will be equal to n)
☗ C(2n,n) is always an even number.
☞ C(2n,n) is the sum of co-efficients in the expansion of (1-x)-n
☻ P(n,r) =r! * C(n,r) which implies P(2n,n)  = (2n)! / n! or P(2n,n) =n! * C(2n,n) = 2n *(2n-1)(2n-3)(2n-5)...........1 ( no. of terms in red will be equal to n)
♜ We take x= floor[(n-1)/2] , then C(2n,n) = 2(x+1) * [(2n-1)(2n-3)....upto (n+1) or (n+2) terms depending whether n is even or odd . In other words Floor (n/2) terms] / (n-x-1)!
♞ Since C(n,r) =(n/r)*C(n-1,r-1), C(2n,n) =2*C(2n-1,n-1)  and C(2n+1,n+1) = [(2n+1)/(n+1)]*C(2n,n)
☂ C(2n+1,n+1)= [(2+1/n)/(1+1/n)]*2C(2n-1,n-1); when n=1, C(2n+1,n+1) = 3*C(2n-1,n-1); As n -->∞ , C(2n+1,n+1) = 4*C(2n-1,n-1);
   
   

 

Computation of C(2n,n) by 3 Different Methods
  n
  2n
Method-I C(2n,n)=(2n)! / (n!)2
A C(2n-1,n-1)
B C(2n+1,n+1)
B/A C(2n+1,n+1) / C(2n-1,n-1)
   
  2n
  (2n-1)(2n-3)(2n-5).........1
  n!
Method-II C(2n,n)=2n *(2n-1)(2n-3)(2n-5)...........1 / n!
     
  x
  2(x+1)
  No. of terms-g
  (2n-1)=g1
  (2n-1)(2n-3)..upto g terms i.e.(n+1) or (n+2) terms
  (n-x-1)!
Method-III C(2n,n) = 2(x+1) * [(2n-1)(2n-3)....upto (n+1) or (n+2) ]/  (n-x-1)!
     
  P(2n,n) = n! * C(2n,n)