1.

Find the value of n such that ` "(i) "^(n)P_(5) =42 xx ""^(n)P_(3), n gt 4 " (ii) "(""^(n)P_(4))/(" "^(n-1)P_(4))=(5)/(3), n gt 4.`

Answer» We know that ` ""^(n)P_(r)=n(n-1)(n-2)...(n-r+1).`
(i) We have
`( ""^(n)P_(5))/(""^(n)P_(3))=42 rArr (n(n-1)(n-2)(n-3)(n-4))/(n(n-1)(n-2))=42`
`rArr(n-3)(n-4)=42. " " ["as "n(n-1)(n-2) ne 0]`
`rArr n^(2)-7n-30=0 rArr (n-10)(n+3)=0 `
` rArr n=10 " "[ because n ne -3," as " n " cannot be negative"]. `
Hence,` n = 10. `
` (ii) ( ""^(n)P_(4))/(""^(n-1)P_(4)) = (5)/(3) rArr (n(n-1)(n-2)(n-3))/((n-1)(n-2)(n-3)(n-4))=(5)/(3) `
`rArr 3n=5(n-4) " " [ "as " (n-1)(n-2)(n-3)(n-4) ne 0] `
` rArr 2n=20 rArr n=10. `
Hence, `n=10. `


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