Kenneth Rosen Edition 6th Exercise 5.5 Example 10 (Page No. 377)

Question

How many ways are there to put four different employees into three indistinguishable offices when each office can contain any number of employees?

Comments

Use multinomial theorem for identical objects.

number of ways of distributing n identical objects into r persons= Combination of like objects

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we know distinguishable vs distinguishable - starling method is applicable

Is for distinguishable vs indistinguishable also - Starling method applicable?

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Starling method is used when there are different objects and identical boxes.

Answer 1

Here offices are identical 

We can put 4 employees into 1 office in 1 way  ⇒  [ $^4C_4$ = 1]

OR

We can put 3 employees into 1 office & 1 employee into another office in 4 ways ⇒  [ $^4C_3 $ = 4]

OR

We can put 2 employees into 1 office and 1 employee into another office and remaining 1 employee into another office in 6 ways ⇒  [ $^4C_2$ = 6]

OR 

We can put 2 employees into 1 office and 2 employees into another office in 3 ways ⇒  [ $\dfrac{^4C_2}{2}$ = $\dfrac{6}{2}$ = 3]

So, total no. of ways = 1 + 4 + 6 + 3 = 14 ways

Comments

should not the answer  be 3^(4)=81

each different person has 3 options as it can go to any of three offices

or can we look at this problem of partitioning of sum 4 in different ways

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Answer is 81 if offices are distinguishable

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we know distinguishable vs distinguishable - starling method is applicable

Is for distinguishable vs indistinguishable also - Starling method applicable?

Answer 2

14 will be the answer

Answer 3

This  question can be considered as division of 4 distinct items into three groups.

one way of grouping - {4},{0},{0} no of ways= 1

second way - {3},{1},{0} no of ways=4!/3!

third way - {2},{2},{0} no of ways=4!/2!2!*1/2!

fourth way={1},{1},{2} no of ways=4!/2!2!

Total ways=14