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