Prove that `C_0C_r+C_1 C_(r+1)+ C_2 C_(r+2)+...............+c_(n-r) C_n=((2n)!)/((n-r)!(n+r)!)`

Explanation

i.e., ` r - 0 = r + 1 - 1 = r + 2 - 2 = …= n -(n-r) = r ` Given ,
` (1 + x)^(n) = C_(0) + C_(1) x C_(2) x^(2) + …+ C_(n-r) x^(n-r) + …+ C_(n) x^(n)` …(i)
Now ,
` (x + 1)^(n) = C_(0) x^(n) + C_(1) x^(n-1) + C_(2) x^(n-2) + ...+ C_(r) x^(n-r) + C_(r+1) x^(n-r-1) + C_(r+2) x^(n-r-2) + ...C_(n)` ...(ii)
On multiplying Eqs.(i) and (ii) , we get
` (1 +x)^(2n) = (C_(0) + C_(1) x + C_(2)x^(2) + ... + C_(n-r) x^(n-r) + ...+ C_(n) x^(n)) xx(C_(0) x^(n) + C_(1) x^(n-1)`
`+ C_(2) x^(n-2) + ...+ C_(r) x^(n-r) + C_(r+1)x^(n-r-1)`
` + C_(r+2) x^(n-r-2) + ...+ C_(n))` ...(iii)
Now , coefficient of `x^(n-r)` on LHS of Eq .(iii) ` = ""^(2n)C_(n-r)`
` = (2n!)/((n-r)!(n+r)!)`
and coefficient of ` x^(n-r)` on RHS of Eq .(iii)
`= C_(0) C_(r) + C_(1) C_(r+1) + C_(2) C_(r+2) + ...+ C_(n-r) C_(n)`
But Eq.(iii) is an identity , therefore cefficient of ` x^(n-r)` in
RHS = coefficient of `x^(n-r)` in LHS
` rArr C_(0) C_(r) + C_(1) C_(r+1) + C_(2) C_(r+2) + ...+ C_(n-r) C_(n)`
` = (2n!)/((n-r)!(n+r)!)`
Aliter Given ,
` (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + ...+ C_(r) x^(r) + C_(r+1) x^(r+1) + C_(r + 2)x^(r+2) + ...+ C_(n-r) x^(r) + ...+ C_(n) x^(n) `...(i) Now , `(1+(1)/(x))^(n) = C_(0) + (C_(1))/(x) + (C_(2))/(x^(2)) + ...+ (C_(r))/(x^(r))+ (C_(r +1))/(x^(r+1)) + (C_(r +2))/(x^(r+2))+ ...+ (C_(n-r))/(x^(n-r) ) +...+ (C_(n))/(x^(n))` ...(ii)
On multiplying Eqs.(i) and (ii) , we get
`((1 +x)^(2n))/(x^(n)) = (C_(0) + C_(1)x + C_(2) x^(2) + ...+ C_(r) x^(r) + C_(r+1) x^(n-r) + ... + C_(n) x^(n))`
`xx(C_(0) + (C_(1))/(x) + (C_(2))/(x^(2)) + ...+ (C_(r))/(x^(r)) + (C_(r +1))/(x^(r +1)) + (C_(r+2))/(x^(r +2)) + ...+ (C_(n-r))/(x^(n-r)) + ...+ (C_(n))/(x^(n)))`...(iii)
Now , coefficient of `(1)/(x^(r))` in RHS
`(C_(0) C_(r) + C_(1)C_(r+1) + C_(2) C_(r +2) + ...+ C_(n-r)C_(n))`
` therefore ` Coefficient of `(1)/(x^(r))` in LHS = Coefficient of ` x^(n-r) ` in
But Eq.(iii) is an identity , therefore ceofficients of `(1)/(x^(r))` in
` rArr C_(0) C_(r) + C_(1) C_(r +1) + C_(2) C_(r+2) + ...+ C_(n-r)C_(n)`
` = (2n!)/((n-r)(n+r)!)`
Corollary I For r = 0
` C_(0)^(2) + C_(1)x^(2) + C_(2) x^(2) + ...+ C_(n)^(2)= (2n!)/((n!)^(2))`
Corollary II for r = 1
`C_(0)C_(1) + C_(1) C_(2) + C_(2) C_(3) + ...+ C_(n-1) C_(n) =(2n!)/((n-1)!(n+1)!)` .
Corollary III For r = 2
`C_(0)C_(1) + C_(1) C_(3) + C_(2) C_(4) + ...+ C_(n-2) C_(n)= (2n!)/((n-2)!(n+2)!)` .