kenneth rosen chapter 5 exercise 5.5 ques 51

Question

How many ways are there to distribute six distinguishable objects into four indistinguishable objects so that each of the boxes contain at least one object??

Plss tell how to solve questions based on distributing  Distinguishable objects into Indistinguishable boxes and Indistinguishable objects into indistinguishable boxes. I am not able to solve problem based on these.

Comments

in this way you can also try

           6 objects can be distributed to 4 indistinguishable object as--(1,1,1,3) &(1,1,2,2)

  Now total possibility=6!/1!*1!*1!*3!*3!+6!/1!*1!*2!*2!*2!*2!=65

l

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Excellent thread for Sterling number. Even sterling number can be visualized as grouping and distribution. Here grouping will be (3,1,1,1) and (2,2,1,1). Now for each grouping find out number of such groups 6!/3! 3! + 6!/ 2!.2!. 2! .2! =65

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@tusharp

I havenot got. Can u explain in a detail??

Answer 1

$\color{red}{Distinguishable \ Objects \ and \ Indistinguishable \ Boxes - }$
It's similar to the problem of - Number of ways to partition a set of r objects into n non-empty subsets.
It is given by $\color{green}{Stirling \ Number \ of \ 2^{nd} \ kind}-  \color{magenta}{S(r,n)} $
Recurrence Equation - $S(r+1,n)=S(r,n-1)+ n.S(r,n)$

For the above question - r =6 and n = 4.
S(6,4) -
1
1    1
1    3     1
1    7     6     1
1    15  25  10    1
1    31  90  $\color{Orange}{65}$    15   1

$\color{DarkBlue}{Note} - 65$ is obtained as $25 \ +\ 4*(10.)$ In the similar way, all the numbers are obtained.
For more detail  - Refer this

$\color{red}{Indistinguishable \ Objects \ and \ Indistinguishable \ Boxes - }$
Refer this.

Comments

It's $S(2,1).$

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yes,I got it now :)

I done an example like 4 distinguishable object and 2 indistinguishable boxes

then if objects are $\left \{ a,b,c,d \right \}$

then no of ways will be

$\left \{ a,b,c \right \} ,\left \{ d \right \}$

$\left \{ a,b,d\right \} ,\left \{ c \right \}$

$\left \{ a,c,d\right \} ,\left \{ b \right \}$

$\left \{ b,c,d\right \} ,\left \{ a \right \}$

$\left \{ a,b\right \} ,\left \{ c,d \right \}$

$\left \{ a,c\right \} ,\left \{ b,d \right \}$

$\left \{ b,c\right \} ,\left \{ a,d \right \}$

And total is 7 ways according to triangle too

right?

thanks a lot :)

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@srestha mam

@Soumya29 mam

is this problem can not be solved without using stirling numbers of the second kind,is there any alternative approach?