If the number of positive integral solutions of `u+v+w=n` be denoted by `P_(n)` then the absolute value of `|{:(P_(n),P_(n+1),P_(n+2)),(P_(n+1),P_(n+2

Monjur Alahi

If the number of positive integral solutions of `u+v+w=n` be denoted by `P_(n)` then the absolute value of `|{:(P_(n),P_(n+1),P_(n+2)),(P_(n+1),P_(n+2),P_(n+3)),(P_(n+2),P_(n+3),P_(n+4)):}|` is
A. `-1`
B. `2`
C. `3`
D. `4`


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Salim Ahmed Kawsar

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Correct Answer - A
`(a)` As `u+v+w=n` and `u,v,w ge 1`
Now, number of solutions of `u+v+w=nimpliesP_(n)=^(n-1)C_(n-3)`
Similarly `P_(n+1)=^(n)C_(n-2)`, `P_(n+2)=^(n+1)C_(n-1)`, `P_(n+3)=^(n+2)C_(n)`, `P_(n+4)=^(n+3)C_(n+1)`.
Now `Delta=|{:(.^(n-1)C_(n-3),.^(n)C_(n-2),.^(n+1)C_(n-1)),(.^(n)C_(n-2),.^(n+1)C_(n-1),.^(n+2)C_(n)),(.^(n+1)C_(n-1),.^(n+2)C_(n),.^(n+3)C_(n+1)):}|`
`=(1)/(8)|{:(((n-1)!)/((n-3)!),(n!)/((n-2)!),((n+1)!)/((n-1)!)),((n!)/((n-2)!),((n+1)!)/((n-1)!),((n+2)!)/(n!)),(((n+1)!)/((n-1)!),((n+2)!)/(n!),((n+3)!)/((n+1)!)):}|`
`=(1)/(8)|{:((n-1)(n-2),n(n-1),n(n+1)),(n(n-1),n(n+1),(n+1)(n+2)),(n(n+1),(n+2)(n+1),(n+3)(n+2)):}|`
`Delta=(1)/(2)|{:(1,n(n-1),n),(1,n(n+1),(n+1)),(1,(n+2)(n+1),(n+2)):}|`
(On applying (first) `C_(3) to C_(3)-C_(2)` and `C_(1)toC_(1)-C_(2)` (and then) `C_(1)toC_(1)+C_(3)`)
`=(1)/(2)|{:(1,n(n-1),n),(0,2n,1),(0,2(n+1),1):}|(R_(3)toR_(3)-R_(2) and R_(2)toR_(2)-R_(1))impliesDelta=-1`

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