1.

If ` ""^(n)P_(4) =2xx""^(5)P_(3)`, find n.

Answer» We have
`""^(n)P_(4)=2xx""^(5)P_(3)`
`rArr n(n-1)(n-2)(n-3)=2xx5xx4xx3`
`rArr n(n-3)(n-1)(n-2)=120`
`rArr (n^(2)-3n)(n^(2)-3n+2)=120`
`rArr m(m+2)-120=0, " where " m=n^(2)-3n`
`rArr m^(2)+2m-120=0rArr (m+12)(m-10)=0`
`rArr m= -12 or m =10.`
This gives
`n^(2)-3n= -12 or n^(2)-3n=10`
`rArr n^(2)-3n+12 =0 or n^(2)-3n-10=0`
`rArr n=(3pm sqrt(9-48))/(2) or (n-5)(n+2)=0`
`rArr n=(3pm isqrt(39))/(2) or n = 5or n = -2`
`rArr n =5 " " `[neglecting the negative and imaginary values of n].
Hence, `n =5.`


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