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?

mathmathj28 Dec 31, 2019

edited by
mathmathj28
Dec 31, 2019

edited by mathmathj28 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

CPhill 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 ] =

n^{k} = 4^{4} = 256 ways

CPhill 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

CPhill Dec 31, 2019