Prove that: `1+^3C_1+^4C_2=^5C_3`

Prove that: .^2C_1+^3C_1+^4C_1=^5C_3-1

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 (i) C_(1)+2C_(2)+3C_(3)+……+nC_(n)=n.2^(n-1) (ii) C_(0)+(C_(1)/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)

Prove that ^m C_1^n C_m-^m C_2^(2n)C_m+^m C_3^(3n)C_m-=(-1)^(m-1)n^mdot

Prove that (C_1)/1-(C_2)/2+(C_3)/3-(C_4)/4++((-1)^(n-1))/n C_n=1+1/2+1/3++1/ndot

Prove that (C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n) = (C_0C_1C_2.....C_(n-1)(n+1)^n)/(n!)

Prove that C_0 + 2.C_1 + 4.C_2 + 8.C_3 + ……+2^n.C_n = 3^n

With usual notations prove that C_0 + 3.C_1 + 3^2.C_2 + ………..+3^n .C_n = 4^n

Prove that 3.C_0 + 6.C_1 + 12.C_2 + ………+3.2^n.C_n = 3^(n+1)

Prove that C_0.C_3 + C_1.C_4 + C_2.C_5 + …..+C_(n-3).C_n = ""^(2n)C_(n +3)