+0  
 
-1
46
4
avatar

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

 

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

 

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

 

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

 Jan 1, 2020
 #1
avatar+106519 
+1

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

 

nk  = 35  =  243  ways

 

 

cool cool cool

 Jan 1, 2020
 #2
avatar+106519 
+1

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 ]

 

 

 

cool cool cool

 Jan 1, 2020
 #3
avatar+106519 
+1

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

 

 

cool cool cool

 Jan 1, 2020
 #4
avatar+106519 
+1

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 

 

cool cool cool

 Jan 1, 2020

18 Online Users

avatar