Question

The number of ways in which $32$ identical blankets can be distributed among $4$ persons so that each of them gets number of blankets which is divisible by $2$ but not by $4,$ is

Answer 1

Each one should get $(4\lambda +2),\lambda \in W$ blankets.
$(4{\lambda }_1+2)+(4{\lambda }_2+2)+(4{\lambda }_3+2)+(4{\lambda }_4+2)=32$
$\Rightarrow 4({\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4)+8=32$
$\Rightarrow {\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4=\frac{24}{4}=6$
Number of ways $={}^{6+4-1}{C}_{4-1}={}^9{C}_3=84$