P&C : 101

NUMBER OF WAYS OF GROUPING SIMILAR THINGS 

In how many ways can you divide five similar objects (say a, a, a, a, a) into three different groups? When we make groups of similar objects, the only way we can differentiate two different ways of grouping is by differentiating between groups when they have different size (number of objects). Therefore, in case of similar objects, the number of different ways we can group them is the number of different sizes of groups that we can make. Let’s see how many groupings of different sizes we can make for 5 similar objects.

 

In my opinion,The  number of ways of grouping 5 similar things to three different groups should be  21 as explained below:
If there are three different groups A, B and C say then the following cases can be made:
                  

Groupings	Number of ways
0, 0, 5	3!/2! ways = 3
0,1,4	3! ways = 6
0,2,3	3! ways = 6
1,1,3	3!/2! ways = 3
1,2,2	3!/2! ways = 3
Total	21 ways