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Let n and k be positive integers such that $n \ge \frac{{k(k + 1)}}{2}$. The number of solutions $({x_1},{x_2},....{x_k})$, ${x_1} \ge 1,{x_2} \ge 2,....{x_k} \ge k,$ all integers, satisfying ${x_1} + {x_2} + .... + {x_k} = n$, is
$^m{C_{k - 1}}$
$^m{C_{k + 1}}$
$^m{C_k}$
None of these {Where $m = \frac{1}{2}(2n - {k^2} + k - 2)$}
Solution
(a) The number of solutions of ${x_1} + {x_2} + … + {x_k} = n$
= Coefficient of tn in $(t + {t^2} + {t^3} + …)({t^2} + {t^3} + …)$
…..$({t^k} + {t^{k + 1}} + ….)$
= Coefficient of ${t^n}$in ${t^{1 + 2 + … + k}}{(1 + t + {t^2} + …)^k}$
But $1 + 2 + … + k = \frac{1}{2}k(k + 1) = r$(say)
and $1 + t + {t^2} + …. = \frac{1}{{(1 – t)}}$
Thus number of required solutions
= Coefficient of ${t^{n – r}}$ in ${(1 – t)^{ – k}}$
=Coefficient of ${t^{n – r}}$ in
$[1 + {\,^k}{C_1}t + {\,^{k + 1}}{C_2}{t^2} + {\,^{k + 2}}{C_3}{t^3} + …]$
$ = {\,^{k + n – r – 1}}{C_{k – 1}}$ ${x^2} – (p + \alpha )x + p\alpha + q\beta = 0$, (where
$m = k + n – r – 1$)
$ = k + n – 1 – \frac{1}{2}k(k + 1) = \frac{1}{2}[2k + 2n – 2 – {k^2} – k]$
$ = \frac{1}{2}(2n – {k^2} + k – 2)$
