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Prove that : (i) (n!)/(r!)=n(n-1)(n-2...

Prove that :
(i) ` (n!)/(r!)=n(n-1)(n-2)...(r+1)`
(ii) `(n-r+1)*(n!)/((n-r+1)!)=(n!)/((n-r)!)`
(iii) `(n!)/(r!(n-r)!)+(n!)/((r-1)!(n-r+1)!)=((n+1)!)/(r!(n-r+1)!)`

Text Solution

Verified by Experts

(i) `RHS =(n(n-1)(n-2)...(r+1)xx(r!))/(r!)=(n!)/(r!).`
(ii) We know that `(n-r+1)! =(n-r+1)*(n-r)!.`
(iii) `LHS=(n!)/((n-r)!*(r-1)!){(1)/(r)+(1)/(n-r+1)}`
`=(n!)/((n-r)!*(r-1)!)*((n+1))/(r(n-r+1))=((n+1)!)/((r!)*(n-r+1)!)=RHS.`
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