Question

Number of ways in which $25$ identical things can be distributed among five persons if each gets odd number of things is

• A. ${}^{25}{C}_4$
• B. ${}^{12}{C}_8$
• C. ${}^{13}{C}_3$
• D. ${}^{14}{C}_4$

Answer 1

The correct option is D ${}^{14}{C}_4$

Let person $P_i$ gets $x_i$ number of things such that :
$x_1+x_2+x_3+x_4+x_5=25$
Let $x_i=2y_i+1$ where $y_i\ge 0$, then
$2(y_1+y_2+y_3+y_4+y_5)+5=25$
$\Rightarrow (y_1+y_2+y_3+y_4+y_5)=10$
The total number of ways is equal to
number of non-negative integral solutions of this equation
$={}^{10+5-1}{C}_{5-1}={}^{14}{C}_4$