Question

The number of ways to distribute 32 different things equally among 4 persons is

• A. $\frac{32!}{8{!}^3}$
• B. $\frac{32!}{8{!}^4}$
• C. $\frac{1}{4}(32!)$
• D. None of these

Answer 1

The correct option is B $\frac{32!}{8{!}^4}$

$3^2$ distinct thing are equally distributed among $4$ people$.$

So each person will get $\frac{32}{4}=8$ things

The $8$ things can be distributed among $4$ people in$:$

The $1^{st}$ person will get $8$ things from remaining $24$ things in ${}^{24}{C}_8$ ways$.$

Similarly $3^{rd}$ person will get $8$ things from remaining $16$ in ${}^{16}{C}_8$ ways$.$

And for $4^{th}$ person ${}^8{C}_8$ ways

$\therefore$ Total ${=}^{32}{C}_8{\times }^{24}{C}_8{\times }^{16}{C}_8{\times }^8{C}_8=\frac{32!}{{8!}^4}$