Imagine $12$ identical balls that are placed in $3$ Different boxes. What is the probability that one of the boxes contains exactly $3$ balls?

Solution 1:

When we work over probability it does not matter whether the balls are distinct or identical. Beucause , when we select one ball to place any box , the selected ball become distict , because it is the first ball selected. Did you get the idea ? It is not anymore identical , because it is in our hands to place and it is the first one.It is valid for the rest , for example the second selected ball , it was the same as the others ,but when we take it from among the others , it become suddenly distinguishable , it become distinguishable because , it is "the second selected ball",now we are able to distinguish it !. Now , you have all distinct balls when you place them , because all of have selection order to be distinguished.

When we comes to your question , we have $12$ balls each have different selected order ,i.e they can be distinguished from other by their selection order. Select $3$ of them to satify given condition. These balls can be second selected ball , sixth selected and tenth selected or third selected , fifth selected and seventh selected. So , you just need to select $3$ order among $12$ order by $C(12,3)$