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

Get the answer to your homework problem.

Try Numerade free for 7 days

We don’t have your requested question, but here is a suggested video that might help.

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

If we let s be the number of ways to do this, then the number of ways to distribute 6 distinguishable objects into 4 distinguishable boxes so that no box is empty is given by $4!(s)=24s$,

$\;\;\;$since there are $4!$ ways to label the boxes.

We can also count the number of ways to distribute the objects into 4 distinguishable boxes so that no box is empty using Inclusion-Exclusion:

If $T$ is the set of all distributions, and $A_i$ is the set of distributions with box i empty, for $1\le i\le 4$,

then $\lvert A_1^c\cap\cdots\cap A_4^c\lvert=\lvert T\lvert-\dbinom{4}{1}\lvert A_1\lvert+\dbinom{4}{2}\lvert A_1\cap A_2\lvert-\dbinom{4}{3}\lvert A_1\cap A_2\cap A_3\lvert$

$\hspace{1.3 in}=\displaystyle 4^6-4\cdot3^6+6\cdot2^6-4\cdot1^6=1560$,

so $24s=1560$ and therefore $s=65$.

This answer is $S(6,4)$, a Stirling number of the second kind.