Question

Number of ways of distribution of 37 identical balls among 13 children , if each gets odd number of balls,

• A. ${}^{49}{C}_{12}$
• B. ${}^{24}{C}_{12}$
• C. ${}^{36}{C}_{12}$
• D. ${}^{24}{C}_{13}$

Answer 1

The correct option is B ${}^{24}{C}_{12}$

Non negative integral solution of $x_1+x_2+....x_{13}=k$ are ${}^{k+n-1}{C}_{n-1}$

Let $a_1$ get $2x_1+1$ balls where $x_1$ can be $0,1,\mathrm{2....}$

Similarly $a_2,a_3...a_{13}$

So, $2x_1+1+2x_2+1+2x_3+1+........2x_{13}+1=37$

$\Rightarrow 2x_1+2x_2+.....2x_{13}=24$

$\Rightarrow x_1+x_2+....x_{13}=12$

Number of solution $={}^{12+13-1}{C}_{13-1}={}^{24}{C}_{12}$

Hence the number of ways of distributing such that each gets odd number of balls $={}^{24}{C}_{12}$