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Question Number 134482 by benjo_mathlover last updated on 04/Mar/21
$$ \\ $$How many ways can this be done if you distribute 25 identical pieces of candy among five children?
Commented by mr W last updated on 04/Mar/21
$${you}'{ll}\:{get}\:{problem}\:{if}\:{you}\:{don}'{t}\:{give} \\ $$$$\left.{each}\:{child}\:\mathrm{5}\:{pieces}!\::\right) \\ $$$$ \\ $$$${mathematically}\:{to}\:{get}\:{an}\:{unique} \\ $$$${answer}\:{you}\:{must}\:{specify}\:{if}\:{each} \\ $$$${child}\:{may}\:{get}\:{nothing}\:{or}\:{must}\:{get} \\ $$$${at}\:{least}\:{one}\:{piece}.\:{if}\:{i}\:{were}\:{you},\:{i}'{ll} \\ $$$${add}\:{if}\:{each}\:{child}\:{must}\:{get}\:{at}\:{least} \\ $$$${two}\:{pieces}. \\ $$
Commented by benjo_mathlover last updated on 04/Mar/21
$$ \\ $$hello sir, that's the problem written in the book no other explanation
Commented by mr W last updated on 04/Mar/21
$${then}\:{it}'{s}\:{assumed}\:{that}\:{a}\:{box}\:\left({here}\:{a}\right. \\ $$$$\left.{child}\right)\:{may}\:{be}\:{empty}.\:{in}\:{this}\:{case} \\ $$$${there}\:{are}\:{C}_{\mathrm{4}} ^{\mathrm{25}+\mathrm{4}} =\frac{\mathrm{29}!}{\mathrm{25}!\mathrm{4}!}=\mathrm{23751}\:{ways}. \\ $$
Commented by mr W last updated on 04/Mar/21
$${if}\:{each}\:{child}\:{must}\:{get}\:{at}\:{least} \\ $$$${two}\:{pieces},\:{then}\:{we}\:{have}\:{in}\:{fact}\:{only} \\ $$$$\mathrm{15}\:{pieces}\:{to}\:{distribute},\:{there}\:{are} \\ $$$${C}_{\mathrm{4}} ^{\mathrm{15}+\mathrm{4}} =\mathrm{3876}\:{ways}. \\ $$
Commented by benjo_mathlover last updated on 04/Mar/21
$$\mathrm{ok}\:\mathrm{sir}.\:\mathrm{thanks} \\ $$