Awesomeee here,

How many ways are there to put 4 distinguishable b***s into 2 indistinguishable boxes?

 

How many ways are there to put some none or all of the b***s into 1 box?  Then I would halve the answer.

 

zero b***s = 1 way

1 ball = 4 ways

2 b***s = 4C2= 6 ways

3 b***s = 4 ways

4 b***s = 1 way

 

1+4+6+4+1=16 ways.

halve it

8 ways  

Question:  How many ways are there to put 4 distinguishable b***s into 2 indtinguishable boxes?

 

Another way to phrase the question is:  How many ways are there to split 4 distinguishable b***s into 2 groups?

 

Label the b***s, A, B, C, and D.

 

You could put all the b***s into one box and leave the other box empty   --->  there is only one way to do this.

 

You could put one ball into one box and the other three b***s into the other box   --->   there are 4 ways to do this

   --->   A & BCD, B & ACD, C & ABD, D & ABC.

 

You could put two b***s into one box and the other two b***s into the other box   --->  there are 3 ways to do this

   --->   AB & CD, AC & BD, AD & BC

 

You could put three b***s into one box and the other ball into the other box   --->   this is a repeat of putting one ball into one box and the other three b***s into the other box

 

So, there is a total of 1 + 4 + 3  =  8 ways.