Question

The number of ways of choosing $10$ balls from infinitely many white, red ,blue and green balls is:

• A. $70$
• B. $84$
• C. $286$
• D. $86$

Answer 1

The correct option is C $286$

The word infinite is just to confuse here.

All we need is $10$ red balls, $10$ white balls $10$ green balls and $10$ black balls.

$\because$ at the end we only need to select $10$ balls.

We want integral solutions of

$Red+White+Green+Black=10$ Where $0\le Red,White,Green,Black\le 10$

We know that integral solutions of $x_1+x_2+....+x_r=n$ is

${}^{n+r-1}{C}_{r-1}$ where $0\le x_1,x_2,x_3,...,x_4\le n$

Hence number of integral solutions of $Red+White+Green$

${=}^{10+4-1}{C}_{4-1}$

${=}^{13}{C}_3$

$=\frac{13\times 12\times 11}{3\times 2\times 1}=13\times 2\times 11=286$