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1. How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable and the boxes are distinguishable?

Let  the number  of balls  =  k

Let the number of boxes  = n

The total number of ways   [ without restriction.....we can have empty boxes ]  is  given by

n^k  = 3⁵  =  243  ways

2. How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable and neither are the boxes?

We  have

5    0    0

4    1    0

3    2    0

3    1    1

2    2    1

5  ways   

   [  The boxes are indistinguishable  so   5  0   0    =    0  5   0   =    0 0 5 ]

3. How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable but the boxes are?

Let  n  =  number  of boxes

Let  k  =  number of balls

The number of  ways  [ with no restrictions....boxes may be empty]   is given by

C( k + n - 1 , n - 1)   =   C( 3 + 5 - 1 , 3 - 1)   =  C( 7, 2)   =  21 ways

4. How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes are not?

Assuming no restrictions  [ boxes may be empty ]  ..... we can compute  this with something known as "Stirling Numbers of the Second Kind"

We have

S(5, 1)  +  S(5,2)  + S (5,3)  =

    1      +    15       +     25     =

              41  ways