There are two urns. Urn A has $$3$$ distinct red balls and urn B has $$9$$ distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

Let the three distinct red balls in urn A be $$R_1 , R_2 , R_3$$ and the nine distinct blue balls in urn B be $$B_1 , B_2 , \dots , B_9$$.

Step 1 (choose from urn A):
We must remove exactly two of the three red balls.
Number of ways $$= {}^{3}C_{2} = 3$$.

Step 2 (choose from urn B):
We must remove exactly two of the nine blue balls.
Number of ways $$= {}^{9}C_{2} = 36$$.

Step 3 (transfer):
The two chosen red balls are sent to urn B and the two chosen blue balls are sent to urn A.
There is no further choice involved in the transfer; once the selections are fixed, the transfer is automatic.

Total number of ways $$= 3 \times 36 = 108$$.

Option C which is: 108