Question

There are two urns.

Urn A has $3$ distinct red balls and urn B has $9$ distinct blue balls.

From each urn, two balls can be taken at random and then transferred to other.

The number of ways in which this can be done.

Answer 1

Solve for the required number of ways:

Given that,

Number of red balls in Urn A $=3$

Number of blue balls in Urn B $=9$

We know that number of ways $r$ objects can be chosen from $n$ objects is ${}^{n}C_r$

Therefore,

The number of ways in which $2$ balls from urn A can be chosen is ${}^{3}C_2$

The number of ways in which $2$ balls from urn B can be chosen is ${}^{9}C_2$

Thus, the number of ways in which $2$ balls from Urn $A$ and $B$ at random $=$${}^{3}C_2$$\times$${}^{9}C_2$

$\begin{array}{clc}=& \frac{3!}{2!\left(3-2\right)!}\times \frac{9!}{2!\left(9-2\right)!}& \left[\because {}^{n}c_r=\frac{n!}{r!\left(n-r\right)!}\right]\\ =& \frac{3\times 2!}{2!\times 1}\times \frac{9\times 8\times 7!}{2\times 7!}& \\ =& 3\times 9\times 4& \\ =& 108& \end{array}$

Hence, the required number of ways is $108$