Question

There are unlimited number of Red, White, Blue and Green balls. The balls are all the same except the color of each ball. What are the number of ways in which selection of 10 balls can be made?

Answer 1

Here, we need maximum $10$ Red balls, $10$ White balls, $10$ Blue balls and $10$ Green balls to have all selections of 10 balls.

Now, we need integral solution of

Red + White + Blue + Green = $10.$

We know that the Integral solution of $x_1+x_2+...+x_r=n$ is given by ${}^{n+r-1}{\text{C}}_{r-1}$

Here $n=10,r=4$

$\therefore$Number of ways of selections are ${}^{10+4-1}{\text{C}}_{4-1}$

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

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

= $13\times 2\times 11$

= $286$

$\therefore$ $286$ ways of selection of $10$ balls can be made.