Show that ` ((2n)!)/(n!) = 2^(n) { 1,3,5 ,…( 2n -1) }`

Prove that ((2n)!)/(n!) =2^(n) (1,3,5,……..(2n-1)) .

Prove that .^(2n)C_(n) = ( 2^(n) xx 1 xx 3 xx …(2n-1))/(n!)

Prove that: ((2n)!)/(n !)={1. 3. 5 (2n-1)}2^ndot

If (1+x)^n=sum_(r=0)^n^n C_r , show that C_0+(C_1)/2++(C_n)/(n+1)=(2^(n+1)-1)/(n+1) .

Show that (1 xx 2^(2) + 2 xx 3^(2) + ... +n xx (n + 1)^(2))/(1^(2) xx 2 + 2^(2) xx 3+ ... + n^(2) xx (n + 1)) = (3n + 5)/(3n + 1)

Prove that .^(n)C_(1) + 2 .^(n)C_(2) + 3 .^(n)C_(3) + "…." + n . ^(n)C_(n) = n 2^(n-1) .

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_m-2n(m ,n in N) Hence, prove that I_(m , n)=f(x)={((n-1)(n-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))pi/4 when both m and n are even ((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))