n का मान निकालिए, यदि
(i) ` .^(2n)C_(3) : .^(n)C_(2) = 12 : 1`
(ii) ` .^(2n)C_(3) : .^(n)C_(3) = 11 : 1`

Explanation

(i) `.^(2n)C_(3) : .^(n)C_(2) = 12 : 1 `
` rArr ((2n)!)/(3!(2n - 3)!) : (n!)/(2!(n-2)!) = 12 : 1`
` rArr((2n) * (2n - 1)(2n-2))/6 : (n(n-1))/2 = 12 : 1`
` rArr ((2n)(2n-1)(2n-2))/6 xx 2/(n(n-1)) = 12/1`
` rArr (2(2n-1)*2)/3 = 12 `
` rArr 2n - 1 = 9`
` rArr 2n - 1 = 9`
` rArr n = 5`
(ii) `.^(2n) C_(3) : .^(n)C_(3) = 11 : 1 `
` rArr .^(2n)C_(3) xx 1 = 11 xx .^(n)C_(3) `
` rArr (2n(2n-1)(2n-2))/(3 xx 2 xx 1) = 11 xx (n(n-1)(n-2))/(3 xx 2 xx 1) `
` rArr (2n(2n-1)*2(n-1))/6 xx 6/(n(n-1)(n-2)) = 11 `
` rArr (4(2n-1))/(n-2) = 11`
` rArr 8n - 4= 11n - 22`
` rArr 22 - 4 = 11n - 8n`
` rArr 18 = 3n rArr n = 6`