Let we have 20 identical object and we want to distribute to 6 different person , how many ways are there that each person get atleast one object ?
solution : x1 + x2 + x3 +x4 + x5 + x6 = 20
where 1 $\leq x _{i} \leq 20$
we have to find coefficient of $x^{20}$ in $( x^{1} + x^{2} + x^{3 } +------+x^{20} )$$^6$
= $x^6(1 + x + x^{2} ------ +x^{19})$$^6$
now we have to find coefficient of $x^{14}$
$\left [ \frac{1-x ^{20}}{1-x} \right ]^{6}$
= 1$\times (_{14}^{6+14-1}\textrm{C} ) = _{14}^{19}\textrm{C}$ $x^{14}$
= 11628
Answer given = 230229
where iam wrong