Prove that following identities:
`"^(n)C_0 `+` "^(n)C_1` + `"^(n)C_2` + …..+` "^(n)C_n` =`2^n`

If ""^(n)C_(3) + ""^(n)C_4 gt ""^(n+1) C_3 , then.

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

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

Prove that "^(n-1)C_3+ ^(n-1)C_4 > ^nC_3 if n >7.

If "^(2n)C_3 : ^nC_2= 12: 1 , find n .

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

Prove that: sum_(r=0)^(n) 3^( r) ""^(n)C_(r) = 4^(n) .

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