+0

# 1. How many ways are there to put balls in boxes if the balls are distinguishable and the boxes are distinguishable? 2. How

+1
48
4

1. How many ways are there to put 4 balls in 4 boxes if the balls are distinguishable and the boxes are distinguishable?

2. How many ways are there to put 4 balls in 4 boxes if the balls are not distinguishable and neither are the boxes?

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

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

Dec 31, 2019
edited by mathmathj28  Dec 31, 2019
edited by mathmathj28  Dec 31, 2019
edited by mathmathj28  Dec 31, 2019

#1
+1

2. How many ways are there to put 4 balls in 4 boxes if the balls are not distinguishable and neither are the boxes?

This is just the number  of  ways to partition  the  number 4

So  we have

4  0    0

3  1    0

2   2   0

2   1   1

The boxes are indistinguishable  so   any   one of these distributions  is also indistinguishable...in other words

4 0  0      is the same as  0 4 0     or   0 0 4

So....4 ways   Dec 31, 2019
#2
+1

1. How many ways are there to put 4 balls in 4 boxes if the balls are distinguishable and the boxes are distinguishable?

Call the  number of balls  =  k

Call the number of boxes   =  n

The number  of  ways to  distribute  these  [ with no restrictions.....a box may be empty   ]  =

nk  =  44  =  256 ways   Dec 31, 2019
#3
+1

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

Call  k the number of balls

Call  n  the number of boxes

The number of ways  [ with no restrictions....a box may  be empty ]  =

C ( k + n - 1 , n - 1)   =  C ( 4 + 4 - 1 , 4 - 1)   =  C ( 7, 3)   =  35 ways   Dec 31, 2019
#4
0

Thank you so much CPhill!!

Jan 8, 2020