Question

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

• A. ${}^{17}{C}_5$ if each gets odd number of things
• B. ${}^{16}{C}_{11}$ if each gets odd number of things
• C. ${}^{14}{C}_5$ if each gets even number things (excluding $0$)
• D. ${}^{15}{C}_{10}$ if each gets even number things (excluding $0$)

Answer 1

Answer: (A), (C)

The correct options are
A ${}^{14}{C}_5$ if each gets even number things (excluding $0$)
C ${}^{17}{C}_5$ if each gets odd number of things

Let person $P_i$ get $x_i$ number of things such that
$x_1+x_2+x_3+x_4+x_5+x_6=30$
If $x_i$ is odd or $x_i=2{\lambda }_i+1$, where ${\lambda }_i\ge 0$, then
$2({\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4+{\lambda }_5+{\lambda }_6)+6=30$
$\Rightarrow {\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4+{\lambda }_5+{\lambda }_6=12$
Then number of soliutions is ${}^{12+6-1}{C}_{6-1}{=}^{17}{C}_5$. If $x_i$ is even or $x_i$
$=2{\lambda }_p$ where ${\lambda }_i\ge 1$, then
$2({\lambda }_i+{\lambda }_2+{\lambda }_3+{\lambda }_4+{\lambda }_5+{\lambda }_6)=30$
$\Rightarrow {\lambda }_i+{\lambda }_2+{\lambda }_3+{\lambda }_4+{\lambda }_5+{\lambda }_6=15$
Therefore, required number of ways is ${}^{15-1}{C}_{6-1}{=}^{14}{C}_5$.