Question

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

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

Answer 1

Answer: (A), (C)

${}^{14}{C}_5$ if each person gets even number of things(excluding $0$).

Let person $P_i$ gets $x_i$ number of things such that :
$x_1+x_2+x_3+x_4+x_5+x_6=30$

Case I - If $x_i$ is odd or $x_i=2y_i+1$, where $y_i\ge 0$, then
$2(y_1+y_2+y_3+y_4+y_5+y_6)+6=30\Rightarrow y_1+y_2+y_3+y_4+y_5+y_6=12$
Then the number of solutions
$={}^{12+6-1}{C}_{6-1}={}^{17}{C}_5$

Case II - If $x_i$ is even or $x_i=2y_i$,
where $y_i\ge 1$(person getting $0$ items is excluded), then
$2(y_1+y_2+y_3+y_4+y_5+y_6)=30\Rightarrow y_1+y_2+y_3+y_4+y_5+y_6=15$
Then the number of solutions
$={}^{15-1}{C}_{6-1}={}^{14}{C}_5$