Number of ways in which 30 identical things are distributed among six persons is

Let person P_i get x_i number of things such that

${\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3+{\mathrm{x}}_4+{\mathrm{x}}_5+{\mathrm{x}}_6=30$

If x_i is odd or ${\mathrm{x}}_{\mathrm{i}}=2{\mathrm{\lambda }}_{\mathrm{i}}+1,\text{ where }{\mathrm{\lambda }}_{\mathrm{i}}\ge 0$, then

$2\left({\mathrm{\lambda }}_{\mathrm{i}}+{\mathrm{\lambda }}_2+{\mathrm{\lambda }}_3+{\mathrm{\lambda }}_4+{\mathrm{\lambda }}_5+{\mathrm{\lambda }}_6\right)+6=30$

or ${\mathrm{\lambda }}_1+{\mathrm{\lambda }}_2+{\mathrm{\lambda }}_3+{\mathrm{\lambda }}_4+{\mathrm{\lambda }}_5+{\mathrm{\lambda }}_6=12$

Then number of solutions is ${ }^{12+6-1}{\mathrm{C}}_{6-1}{=}^{17}{\mathrm{C}}_5$.  

If x_i is even

or ${\mathrm{x}}_{\mathrm{i}}=2{\mathrm{\lambda }}_{\mathrm{I}}\text{, where }{\mathrm{\lambda }}_{\mathrm{i}}\ge 1$, then

$2\left({\mathrm{\lambda }}_{\mathrm{i}}+{\mathrm{\lambda }}_2+{\mathrm{\lambda }}_3+{\mathrm{\lambda }}_4+{\mathrm{\lambda }}_5+{\mathrm{\lambda }}_6\right)=30$

or ${\mathrm{\lambda }}_{\mathrm{i}}+{\mathrm{\lambda }}_2+{\mathrm{\lambda }}_3+{\mathrm{\lambda }}_4+{\mathrm{\lambda }}_5+{\mathrm{\lambda }}_6=15$

Therefore, required number of ways is ${ }^{15-1}{\mathrm{C}}_{6-1}{=}^{14}{\mathrm{C}}_5$.