Question

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

• A. ${}^{24}{C}_4$
• B. ${}^{12}{C}_8$
• C. ${}^{14}{C}_{10}$
• D. ${}^{13}{C}_3$

Answer 1

The correct option is C ${}^{14}{C}_{10}$

$\text{Concept: Total number of non-negative integral solution of}{x}_1+x_2+......+x_r=n\text{is}{}^{n+r-1}{C}_{r-1}$

Also, $n$ identical things can be distributed in $r$ groups in ${}^{n+r-1}{C}_{r-1}$ ways

Let person $P_i$ gets $x_i$ number of things such that
$x_1+x_2+x_3+x_4+x_5=25$
Let $x_i=2{\lambda }_i+1$, where ${\lambda }_i\ge 0$. Then
$2({\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4+{\lambda }_5)+5=25$
or ${\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4+{\lambda }_5=10$ (i)
Required number of ways $=$ The number of non-negative integral solutions of the equation (i),
which is equal to ${}^{10+5-1}{C}_{5-1}{=}^{14}{C}_4={}^{14}{C}_{10}$.