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 $=\_{}^{4}C_{2}\times \frac{4 !}{2 !}\times \_{}^{3}C_{2}=6\times 12\times 3=216$