Question
The number of positive integral solutions of $$x + y + z = n,n \in N,n \geqslant 3,$$ is
A.
$$^{n - 1}{C_2}$$
B.
$$^{n - 1}{P_2}$$
C.
$$n\left( {n - 1} \right)$$
D.
None of these
Answer :
$$^{n - 1}{C_2}$$
Solution :
The number of positive integral solutions
= co-efficient of $${x^n}\,{\text{in }}{\left( {x + {x^2} + {x^3} + .....} \right)^3}$$
= co-efficient of $${x^{n - 3}}\,{\text{in }}{\left( {1 + x + {x^2} + .....} \right)^3}$$
= co-efficient of $${x^{n - 3}}\,{\text{in }}{\left( {1 - x} \right)^{ - 3}}$$
= co-efficient of $${x^{n - 3}}\,{\text{in }}\left\{ {^2{C_0} + {\,^3}{C_1}x + {\,^4}{C_2}{x^2} + .....} \right\}$$
$$ = {\,^{n - 1}}{C_{n - 3}} = \frac{{\left( {n - 1} \right)!}}{{\left( {n - 3} \right)!\,\,2!}} = \frac{{\left( {n - 1} \right)\left( {n - 2} \right)}}{2}.$$