Number-of-ways-n-differrnt-things-be-distributed-in-r-identical-boxes-so-as-1-empty-box-is-allowed-2-empty-box-not-allowed-Number-of-ways-n-identical-things-be-distributed-to-r-identical-boxes-s

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Question Number 165385 by SLVR last updated on 31/Jan/22

$$Numberofways..n$$$$differrntthingsbedistributed$$$$inridenticalboxessoas$$$$1)emptyboxisallowed$$$$2)emptyboxnotallowed$$$$Numberofways\dots nidentical$$$$thingsbedistributedtor$$$$identicalboxessoas$$$$1)emptyboxallowed$$$$2)emptyboxnotallowed$$$$$$

Commented by SLVR last updated on 31/Jan/22

$$kindly\dots provideproofif$$$$possible$$

Answered by mr W last updated on 01/Feb/22

$$I.$$$$ndifferentobjectsintoridentical$$$$boxes$$$$a)nonemptybox$$$$S(n,r)\leftarrow stirlingnumberofthesecondkind$$$$b)emptyboxallowed$$$$S(n,1)+S(n,2)+\dots +S(n,r)$$$$$$$$II.$$$$nidenticalobjectsintoridentical$$$$a)nonemptybox$$$$p(n,r)\leftarrow numberofpartionsofnintorparts$$$$b)emptyboxallowed$$$$p(n,1)+p(n,2)+\dots +p(n,r)$$

Commented by SLVR last updated on 02/Feb/22

$$Sokindofyousir\dots weareblessed$$$$HereiamnewtoS(n,r)and$$$$itriedwithGooglesearch..$$$$buticouldnotgetthrough..$$$$kidlyexplainS(9,5)andS(5,3)$$$$alsop(n,r)withp(5,3).Kindly$$$$blessmesir\dots$$

Commented by mr W last updated on 02/Feb/22

https://en.m.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind

Commented by mr W last updated on 02/Feb/22

https://en.m.wikipedia.org/wiki/Partition_(number_theory)

Commented by SLVR last updated on 06/Feb/22

$$Sokindofyouprof.W\dots Godblessyou$$

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