Question

The number of ways of distributing $4$ blue balls $5$ yellow balls and $3$ red balls among $4$ children (considering ball of same colour as identical) is

• A. ${}^{11}{C}_3$
• B. ${}^6{C}_3\times {}^7{C}_3\times {}^5{C}_3$
• C. ${}^{15}{C}_3$
• D. ${}^7{C}_3\times {}^8{C}_3\times {}^6{C}_3$

Answer 1

The correct option is D ${}^7{C}_3\times {}^8{C}_3\times {}^6{C}_3$

Let blue balls received by children be $x_1,x_2,x_3,x_4$

For $4$ blue balls
$x_1+x_2+x_3+x_4=4$
Number of ways in which they can receive them is ${}^{4+4-1}{C}_{4-1}={}^7{C}_3$

For $5$ yellow balls
Let yellow balls received by children be $y_1,y_2,y_3,y_4$
$y_1+y_2+y_3+y_4=5$
Number of ways in which they can receive them is${}^{4+5-1}{C}_{4-1}={}^8{C}_3$

For $3$ red balls
Let red balls recieved by children be $z_1,z_2,z_3,z_4$
$z_1+z_2+z_3+z_4=3$
Number of ways in which they can recieve them is ${}^{4+3-1}{C}_{4-1}={}^6{C}_3$

$\therefore$ required ways $={}^7{C}_3\times {}^8{C}_3\times {}^6{C}_3$