JEE Advance - Mathematics (1996 - No. 16)
Let n and k be positive such that $$n \ge {{k(k + 1)} \over 2}$$ . 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......................................
$${{\left[ {k + \left( {n - {{k(k + 1)} \over 2}} \right) - 1} \right]!} \over {\left[ {n - {{k(k + 1)} \over 2}} \right]!\,(k - 1)\,!}}$$
$${{\left[ {k + \left( {n + {{k(k + 1)} \over 2}} \right) - 1} \right]!} \over {\left[ {n - {{k(k + 1)} \over 2}} \right]!\,(k - 1)\,!}}$$
$${{\left[ {k + \left( {n - {{k(k + 1)} \over 2}} \right) - 1} \right]!} \over {\left[ {n - {{k(k + 1)} \over 2}} \right]!\,(k + 1)\,!}}$$
$${{\left[ {k - \left( {n - {{k(k + 1)} \over 2}} \right) - 1} \right]!} \over {\left[ {n - {{k(k + 1)} \over 2}} \right]!\,(k - 1)\,!}}$$
$${{\left[ {k + \left( {n - {{k(k + 1)} \over 2}} \right) + 1} \right]!} \over {\left[ {n - {{k(k + 1)} \over 2}} \right]!\,(k - 1)\,!}}$$
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