prove: `47C_4 +51C_3+50C_3+43C_3+48C_3+47C_3=52C_4`

If C_1,C_2,C_3,C_4 are the coefficients of any four consecutive terms in the expansion of (1+ x)^n , prove that : C_1/(C_1+C_2)+C_3/(C_3+C_4)=(2C_2)/(C_2+C_3) .

Prove that: .^(47)C_(4) + .^(51)C_(3) +^(50)C_(3)+^(49) C_(3) +^(48)C_(3) +^(47)C_(3) =^(52)C_(4)

If C_1, C_2 , C_3 , C_4 are the coefficients of any consecutive terms in the expansion of (1+x)^n , prove that : (C_1)/(C_1+C_2) + (C_3)/(C_3+C_4) = (2C_2)/(C_2 + C_3)

Prove that : ^2C_1+ ^3C_1+ ^4C_1= "^3C_2+ ^4C_2 .

The number of ordered triplets of positive integers which satisfy the inequality 20lex+y+zle50 is (A) ""^50C_3- "^19C_3 (B) ""^50C_2-"^19C_2 (C) ""^51C_3-"^20C_3 (D) none of these

The number of ordered triplets of positive integers which satisfy the inequality 20lex+y+zle50 is (A) ""^50C_3- "^19C_3 (B) ""^50C_2-"^19C_2 (C) ""^51C_3-"^20C_3 (D) none of these

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

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

C_0 + C_1 + 2.C_2(3) + 3.C_3(3^2)+ 4.C_4(3^3) + ……+n.C_n 3^(n-1) =