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Let min N and C(r) = ""^(n)C(r), for 0 ...

Let m`in` N and `C_(r) = ""^(n)C_(r)`, for ` 0 le r len`
Statement-1: `(1)/(m!)C_(0) + (n)/((m +1)!) C_(1) + (n(n-1))/((m +2)!) C_(2) +… + (n(n-1)(n-2)….2.1)/((m+n)!) C_(n)`
` = ((m + n + 1 )(m+n +2)…(m +2n))/((m +n)!)`
Statement-2: For r `le`0
`""^(m)C_(r)""^(n)C_(0)+""^(m)C_(r-1)""^(n)C_(1) + ""^(m)C_(r-2) ""^(n)C_(2) +...+ ""^(m)C_(0)""^(n)C_(r) = ""^(m+n)C_(r)`.


(a) Statement-1 and Statement-2 both are correct and Statement-2 is the correct explanation for the Statement-1.
(b) Statement-1 and Statement-2 both are correct and Statement-2 is not the correct explanation for the Statement-1.
(c) Statement-1 is correct but Statement-2 is wrong.
(d) Statement-2 is correct but Statement-1 is wrong.

Text Solution

Verified by Experts

`1/(m!).^(n)C_(0)+(n)/((m+1)!).^(n)C_(1)+(n(n-1))/((m+2)!).^(n)C_(2)+"....."+(n(n-1)xx"...."xx2xx1)/((m+2)!).^(n)C_(n)`
`= (n!)/((m+n)!)(((m+n)!)/(m!n!).^(n)C_(0)+((m+n)!n)/((m+1)!n!).^(n)C_(1)+((m+n)!n(n-1))/((m+2)!n!) xx .^(n)C_(2)+"....."+((m+n)!)/(n!)+"....."+((m+n)!)/(n!)(n(n-1)xx"....."xx2xx1)/((m+n)!).^(n)C_(n))`
`= (n!)/((m+n)!)(.^(m+n)C_(n).^(n)C_(0)+((m+n)!)/((m+1)!(n-1)!).^(n)C_(1)+((m+n)!)/((m+2)!(n-2)!)xx.^(n)C_(2) + "....."+((m+n)!)/(1)(1)/((m+n)!)(1)/((m+n)!) .^(n)C_(n))`.
`= (n!)/((m+n)!)(.^(m+n)C_(n).^(n)C_(0)+.^(m+n)C_(n-1).^(n)C_(1)+.^(m+n)C_(n-2).^(n)C_(2)+"...."+.^(m+n)C_(0).^(n)C_(n))`
`= (n!)/((m+n)!)`[coefficient of `x^(n)` in `(1+x)^(m+n)(1+x)^(n)`]
`= (n!)/((m+n)!)`[coefficient of `x^(n)` in `(1+x)^(m+2n)`]
`= (n!)/((m+n)!).^(m+2n)C_(n)`
`= (n!)/((m+n)!)((m+2n)!)/((m+n)!n!)`
`= ((m+2n)!)/((m+n)!(m+n)!)`
`= ((m+n+1)(m+n+2)(m+n+3)"...."(m+2n))/((m+n)!)`
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