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Question Number 165385 by SLVR last updated on 31/Jan/22
$${Number}\:{of}\:\:{ways}..{n}\: \\ $$$${differrnt}\:\:{things}\:{be}\:{distributed} \\ $$$${in}\:{r}\:{identical}\:{boxes}\:{so}\:{as} \\ $$$$\left.\mathrm{1}\right){empty}\:{box}\:{is}\:{allowed} \\ $$$$\left.\mathrm{2}\right){empty}\:{box}\:{not}\:{allowed} \\ $$$${Number}\:{of}\:{ways}...{n}\:{identical} \\ $$$${things}\:{be}\:{distributed}\:{to}\:{r} \\ $$$${identical}\:{boxes}\:{so}\:{as} \\ $$$$\left.\mathrm{1}\right){empty}\:{box}\:{allowed} \\ $$$$\left.\mathrm{2}\right){empty}\:{box}\:{not}\:{allowed} \\ $$$$ \\ $$
Commented by SLVR last updated on 31/Jan/22
$${kindly}...{provide}\:{proof}\:\:\:{if} \\ $$$${possible} \\ $$
Answered by mr W last updated on 01/Feb/22
$${I}. \\ $$$${n}\:{different}\:{objects}\:{into}\:{r}\:{identical} \\ $$$${boxes} \\ $$$$\left.{a}\right)\:{non}\:{empty}\:{box}\: \\ $$$${S}\left({n},{r}\right)\:\leftarrow{stirling}\:{number}\:{of}\:{the}\:{second}\:{kind} \\ $$$$\left.{b}\right)\:{empty}\:{box}\:{allowed} \\ $$$${S}\left({n},\mathrm{1}\right)+{S}\left({n},\mathrm{2}\right)+...+{S}\left({n},{r}\right) \\ $$$$ \\ $$$${II}. \\ $$$${n}\:{identical}\:{objects}\:{into}\:{r}\:{identical} \\ $$$$\left.{a}\right)\:{non}\:{empty}\:{box} \\ $$$${p}\left({n},{r}\right)\:\leftarrow{number}\:{of}\:{partions}\:{of}\:{n}\:{into}\:{r}\:{parts} \\ $$$$\left.{b}\right)\:{empty}\:{box}\:{allowed} \\ $$$${p}\left({n},\mathrm{1}\right)+{p}\left({n},\mathrm{2}\right)+...+{p}\left({n},{r}\right) \\ $$
Commented by SLVR last updated on 02/Feb/22
$${So}\:{kind}\:{of}\:{you}\:{sir}...{we}\:{are}\:{blessed} \\ $$$${Here}\:{i}\:{am}\:{new}\:{to}\:{S}\left({n},{r}\right)\:{and}\: \\ $$$${i}\:{tried}\:{with}\:{Google}\:{search}.. \\ $$$${but}\:{i}\:{could}\:{not}\:{get}\:{through}.. \\ $$$${kidly}\:{explain}\:{S}\left(\mathrm{9},\mathrm{5}\right)\:{and}\:{S}\left(\mathrm{5},\mathrm{3}\right) \\ $$$${also}\:{p}\left({n},{r}\right)\:{with}\:{p}\left(\mathrm{5},\mathrm{3}\right).{Kindly} \\ $$$${bless}\:{me}\:{sir}... \\ $$
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
$${So}\:{kind}\:{of}\:{you}\:{prof}.{W}...{God}\:{bless}\:{you} \\ $$