Question

If $25$ blankets are distributed among $5$ persons then the probability that each person gets odd number of blankets is

• A. $\frac{{}^{14}{C}_{10}}{{}^{29}{C}_4}$
• B. $\frac{{}^{17}{C}_4}{{}^{29}{C}_4}$
• C. $\frac{1}{2}$
• D. $\frac{{}^{10}{C}_4}{{}^{29}{C}_4}$

Answer 1

The correct option is A $\frac{{}^{14}{C}_{10}}{{}^{29}{C}_4}$

Number of blankets each person should get is odd so each person can get $2{\lambda }_i+1$ blankets where ${\lambda }_i\ge 0$
For $5$ persons
$\Rightarrow 2({\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4+{\lambda }_5)+5=25$
$\Rightarrow {\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4+{\lambda }_5=10$
Number of ways of distributing odd number of blankets will be
$\Rightarrow {}^{10+5-1}{C}_{5-1}={}^{14}{C}_4={}^{14}{C}_{10}$
Total number of ways to distribute the blankets without any restriction
$\Rightarrow {}^{25+5-1}{C}_{5-1}={}^{29}{C}_4$
$\therefore \text{Required probability}=\frac{{}^{14}{C}_{10}}{{}^{29}{C}_4}$