Find the number of ways to put the balls into the boxes with no restrictions?

Question

Find the number of ways to put the 7 balls (Distinct) into the 3 boxes (Indistinguishable) without any restrictions ?

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Can I apply stirling formula of 2nd kind in this case ?

Answer 1

Yes you can, Stirling numbers of the second kind give you the correct answer for this one:

The Stirling numbers of the second kind, written ${\displaystyle S(n,k)}$ or ${\displaystyle \lbrace \textstyle {n \atop k}\rbrace }$ or with other notations, count the number of ways to partition a set of ${\displaystyle n}$ labelled objects into ${\displaystyle k} $ nonempty unlabelled subsets

You have no restrictions so I assume you can have empty boxes. Therefore the total number of ways will be

n.o of ways to put all balls in 1 box + n.o of ways to put all balls in 2 boxes + n.o of ways to put all balls in 3 boxes.

There is only 1 way of having everything in the same box. Using the stirling numbers, there are $S(7, 2) $ ways of having the balls in 2 different boxes and $S(7, 3) $ ways of having them split between 3 boxes. Now just use the recurrence formula to calculate it.