1.

सिद्ध कीजिए कि- (i) `""^(n)P_(r) = (n-r+1) . ""^(n)P_(r-1)` , (ii) `""^(n)P_(r) = ""^(n-1)P_(r) +r , ""^(n-1)P_(r-1)` (iii) `(""^(n)P_(r))/(""^(n)P_(r-2)) = (n-r+2) (n-r+1)` (iv) `""^(2n)p_(n) = 2^(n)[1.3.5"...."(2n-1)]`

Answer» (i) दायाँ पक्ष `= (n-r+1) .""^(n)P_(r-1)`
`= (n-r+1).(n!)/((n-r+1)!)`
`=(n-r+1).(n!)/((n-r+1)(n-r)!)`
`rArr = (n!)/((n-r)!) = ""^(n)P_(r) =` बायाँ पक्ष
(ii) दायाँ पक्ष `=""^(n-1)P_(r)+r.""^(n-1)P_(r-1)`
`=((n-1)!)/((n-1-r)!) +r.((n-1)!)/((n-1-r+1)!)`
`= (n-1)![(1)/((n-r-1)!) +(r)/((n-r)!)]`
`(n-1)![(n-r)/((n-r)(n-r-1)!)+(r)/((n-r)!)]`
`= (n-1)![(n-r+r)/((n-r)!)]=((n-1)!n)/((n-r)!)`
`= (n!)/((n-r)!) = .^(n)P_(r)=` बायाँ पक्ष
(iii) बायाँ पक्ष `=(""^(n)P_(r))/(""^(n)P_(r-2))`
`=(n!)/((n-r)!) xx ((n-r+2)!)/(n!)`
`= ((n-r+2)!)/((n-r)!) = ((n-r+2)(n-r+1)(n-r)!)/((n-r)!)`
`= (n-r+2)(n-r+1) = ` दायाँ पक्ष
(iv) बायाँ पक्ष `= ""^(2n)p_(n) = (2n!)/((2n-n)!) = (2n!)/(n!)`
`= (1.2.3"..."(2n-2)(2n-1)(2n))/(n!)`
`= ({1.3.5"..."(2n-1)}.{2.4.6.8"...."(2n-2)(2n)})/(n!)`
`=({1.3.5.7 "...."(2n-1)}.2^(n).n!)/(n!)`
`= 2^(n) {1.3.5"..."(2n-1)}`
`=`दायाँ पक्ष


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