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[" Prove that "sum(r=0)^(n)3^(n)C(r)=1^(...

[" Prove that "sum_(r=0)^(n)3^(n)C_(r)=1^(n)],[v^(n)=^(n)C_(n)+^(n)C_(1)x+^(n)C_(2)x^(2)+^(n)C_(3)x^(3)+....+^(n)C_(n)x^(n)]

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If (1-x^(2))^(n)=sum_(r=0)^(n)a_(r)x^(r)(1-x)^(2n-r), then a_(r) is equal to ^(n)C_(r) b.^(n)C_(r)3^(r) c.^(2n)C_(r) d.^(n)C_(r)2^(r)

If sum_(r=0)^(n){("^(n)C_(r-1))/('^(n)C_(r )+^(n)C_(r-1))}^(3)=(25)/(24) , then n is equal to

Prove that sum_(r = 1)^n r^3 ((C_r)/(C_(r - 1))^2) = (n (n + 1)^2 (n+2))/(12)

If (1+2x+x^(2))^(n)=sum_(r=0)^(2n)a_(r)x^(r), then a=(^(n)C_(2))^(2) b.^(n)C_(r).^(n)C_(r+1) c.^(2n)C_(r) d.^(2n)C_(r+1)

If (1+x)^(n)=^(n)C_(0)+^(n)C_(1)x+^(n)C_(2)x^(2)+…+^(n)C_(n)x^(n) , prove that, nC_(1)-2^(n)C_(2)+3^(n)C_(3)-…+(-1)^(n-1).n^(n)C_(n)=0 .

""^(n)C_(n-r)+3.""^(n)C_(n-r+1)+3.""^(n)C_(n-r+2)+""^(n)C_(n-r+3)=""^(x)C_(r)

""^(n)C_(n-r)+3.""^(n)C_(n-r+1)+3.""^(n)C_(n-r+2)+""^(n)C_(n-r+3)=""^(x)C_(r)

""^(n)C_(n-r)+3.""^(n)C_(n-r+1)+3.""^(n)C_(n-r+2)+""^(n)C_(n-r+3)=""^(x)C_(r)

""^(n)C_(n-r)+3.""^(n)C_(n-r+1)+3.""^(n)C_(n-r+2)+""^(n)C_(n-r+3)=""^(x)C_(r)