Find n if `.^(n)C_2 = "^(n)C_6`

Find n, if ""^(2n)C_(1),""^(2n)C_(2) and ""^(2n)C_(3) are in A.P.

Find n if : 30 ^nP_6= "^(n+2)P_7 .

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

Find the value of ^4nC_0+^(4n)C_4+^(4n)C_8+....+""^(4n)C_(4n) .

If ([""^(n)C_(r) + 4*""^(n)C_(r+1) + 6*""^(n)C_(r+2)+ 4*""^(n)C_(r+3) + ""^(n)C_(r+4)])/([""^(n)C_(r) + 3. ""^(n)C_(r+1)+ 3*""^(n)C_(r+2) + ""^(n)C_(r +3)])=(n + lambda)/(r+lambda) the value of lambda is

Prove that "^nC_r+2 ^(n)C_(r-1)+ ^(n)C_(r-2) = ^(n+2)C_r .

If ""^(n)C_(r ): ""^(n)C_(r+1)=1:2 and ""^(n)C_(r+1): ""^(n)C_(r+2)=2:3 , find n and r.

If .^(n)C_(9)=.^(n)C_(7) , find n.

The number of ways of choosing triplet (x , y ,z) such that zgeqmax{x, y}a n dx ,y ,z in {1,2, n ,n+1} is a. ^n+1C_3+^(n+2)C_3 b. n(n+1)(2n+1)//6 c. 1^2+2^2++n^2 d. 2((^(n+2)C_3))_(-^(n+2))C_2