1.

सिद्ध कीजिए कि - (i) `""^(n-1)C_(r-1) + ""^(n-1)C_(r) = ""^(n)C_(r)` , (ii) `""^(n)C_(r ) + ""^(n-1)C_(r-1) + ""^(n-1)C_(r-2) = ""^(n+1)C_(r )` (iii) `""^(n)C_(r-2) + 2.""^(n)C_(r-1) + ""^(n)C_(r) = ""^(n+2)C_(r )`

Answer» बायाँ पक्ष `""^(n-1)C_(r-1) + 1 ""^(n-1)C_(r ) = ((n-1)!)/((r-1)!(n-r)!) + ((n-1)!)/(r!(n-r-1)!)`
`= ((n-1)!)/((r-1)!(n-r)(n-r-1)!) + ((n-1)!)/(r(r-1)!(n-r-1)!)`
`= ((n-1)!)/((r-1)!(n-r-1)!) [1/(n-r) +1/r]`
`= ((n-1)!n)/((r-1)!(n-r-1)!) [(r+n-r)/(r(n-r))]`
`= ((n-1)!n)/(r(r-1)!(n-r)(n-r-1)!) `
`= (n!)/(r!(n-r)!) = ""^(n)C_(r)` दायाँ पक्ष


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