Prove that `C_(0)^(2) + C_(1)^(2) + C_(2)^(2) +…+C_(n)^(2) = (2n!)/(n!)^(2)`

Prove that sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .

If C_(r) = .^(n)C_(r) then prove that (C_(0) + C_(1)) (C_(1) + C_(2)) "….." (C_(n-1) + C_(n)) = (C_(1)C_(2)"…."C_(n-1)C_(n))(n+1)^(n)//n!

Prove that (1^(2))/(3).^(n)C_(1)+(1^(2) + 2^(2))/(7).^(n)C_(2)+(1^(2)+2^(2)+3^(2))/(7).^(n)C_(3)+"...." +(1^(2)+2^(3)+"....."+n^(2))/(2n+1).^(n)C_(n) = (n(n+3))/(6)2^(n-2) .

Prove that .^(n)C_(0) - ^(n)C_(1) + .^(n)C_(2)- ^(n)C_(3) + "…" + (-1)^(r) + .^(n)C_(r) + "…" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that ^n C_0 ^(2n)C_n- ^n C_1 ^(2n-2)C_n+ ^n C_2^(2n-4)C_n-=2^ndot

Prove that ^n C_1(^n C_2)(^n C_3)^3(^n C_n)^nlt=((2^n)/(n+1))^(n+1_C()_2),AAn in Ndot

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

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

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

If C_(r ) = (n!)/(r!(n - r)!) , then prove that sqrt(C_(1)) + sqrt(C_(2)) + …. + sqrt(C_(n)) sqrt(n(2^(n) - 1))