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Determine n if (i) .^(2n)C(3) : .^(n)C...

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

Text Solution

Verified by Experts

(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 n = 5`
(ii) `.^(2n)C_(3):^(n)C_(3) = 11:1`
`rArr .^(2n)C_(3)xx1 = 11 xx .^(n)C_(3)`
`rArr (2n(2n-1)(2n-2))/(3xx 2xx1) = 11 xx (n(m-1)(n-2))/(3 xx 2xx 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 8 n - 4 = 11 n - 22`
`rArr 22 - 4 = 11n - 8n`
`rArr 18 = 3n rArr n = 6`.
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