Determine n if
(i) `.^(2n)C_(3) : .^(n)C_(3) = 12:1`
(ii) `.^(2n)C_(3): .^(n)C_(3)= 11:1`

Determine n if (i) .^(2n) C_2: "^n C_2=12 :1 (ii) .^(2n) C_3: "^n C_3=11 :1

Determine n if (i) ^2n C_2:^n C_2=12 :1 (ii) ^2n C_3:^n C_3=11 :1

Determine n if (i) ^2n C_2:^n C_2=12 :1 (ii) ^2n C_3:^n C_3=11 :1

Determine n if (i) ^2n C_2:^n C_2=12 :1 (ii) ^2n C_3:^n C_3=11 :1

If .^(2n)C_(3):^(n)C_(3)=11:1 , find the value of n.

Prove that: (i) r.^(n)C_(r) =(n-r+1).^(n)C_(r-1) (ii) n.^(n-1)C_(r-1) = (n-r+1) .^(n)C_(r-1) (iii) .^(n)C_(r)+ 2.^(n)C_(r-1) +^(n)C_(r-2) =^(n+2)C_(r) (iv) .^(4n)C_(2n): .^(2n)C_(n) = (1.3.5...(4n-1))/({1.3.5..(2n-1)}^(2))

Find the sum .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "……" + n xx .^(n)C_(n) .

Find the following sums : (i) .^(n)C_(0)-.^(n)C_(2)+.^(n)C_(4)-.^(n)C_(6)+"....." (ii) .^(n)C_(1)-.^(n)C_(3)+.^(n)C_(5)-.^(n)C_(7)+"...." (iii) .^(n)C_(0)+.^(n)C_(4)+.^(n)C_(8)+.^(n)C_(12)+"....." (iv) .^(n)C_(2) + .^(n)C_(6) + .^(n)C_(10)+.^(n)C_(14)+"......" (v) .^(n)C_(1) + .^(n)C_(5)+.^(n)C_(9)+.^(n)C_(13)+"...." (vi) .^(n)C_(3) + .^(n)C_(7) + .^(n)C_(11) + .^(n)C_(15) + "....."

Prove that .^(n-1)C_(3)+.^(n-1)C_(4) gt .^(n)C_(3) if n gt 7 .

The greatest value of n for which the determinant Delta = |(1,1,1),(.^(n)C_(1),.^(n+3)C_(1),.^(n+6)C_(1)),(.^(n)C_(2),.^(n+3)C_(2),.^(n+6)C_(2))| is divisible by 3^(n) , is