Question

The number of ways in which four different toys and five indistinguishable marbles can be distributed between $3$ boys, if each boy receives at least one toy and at least one marble, is

• A. $42$
• B. $100$
• C. $150$
• D. $216$

Answer 1

Answer: (D)

$5$ identical marbles can be distributed to $3$ boys, each receiving at least one in ${}^{5-1}{C}_{3-1}=\_{}^4{C}_2$ ways
$4$ different toys can be grouped as $2,1,1$ in $\frac{4!}{2!1!1!}=\frac{4!}{2!}$ ways
Two boys will get $1$ each and they can be selected in $\_{}^3{C}_2$ ways.
Required number of ways $=\_{}^4{C}_2\times \frac{4!}{2!}\times \_{}^3{C}_2=6\times 12\times 3=216$