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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?\\n

Commented bymr W last updated on 04/Mar/21

you′ll get problem if you don′t give  each child 5 pieces! :)    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.

$${you}'{ll}\:{get}\:{problem}\:{if}\:{you}\:{don}'{t}\:{give} \\ $$ $$\left.{each}\:{child}\:\mathrm{5}\:{pieces}!\::\right) \\ $$ $$ \\ $$ $${mathematically}\:{to}\:{get}\:{an}\:{unique} \\ $$ $${answer}\:{you}\:{must}\:{specify}\:{if}\:{each} \\ $$ $${child}\:{may}\:{get}\:{nothing}\:{or}\:{must}\:{get} \\ $$ $${at}\:{least}\:{one}\:{piece}.\:{if}\:{i}\:{were}\:{you},\:{i}'{ll} \\ $$ $${add}\color{mathred}{\:}{\color{mathred}{i}\color{mathred}{f}}\color{mathred}{\:}{\color{mathred}{e}\color{mathred}{a}\color{mathred}{c}\color{mathred}{h}}\color{mathred}{\:}{\color{mathred}{c}\color{mathred}{h}\color{mathred}{i}\color{mathred}{l}\color{mathred}{d}}\color{mathred}{\:}{\color{mathred}{m}\color{mathred}{u}\color{mathred}{s}\color{mathred}{t}}\color{mathred}{\:}{\color{mathred}{g}\color{mathred}{e}\color{mathred}{t}}\color{mathred}{\:}{\color{mathred}{a}\color{mathred}{t}}\color{mathred}{\:}{\color{mathred}{l}\color{mathred}{e}\color{mathred}{a}\color{mathred}{s}\color{mathred}{t}} \\ $$ $${\color{mathred}{t}\color{mathred}{w}\color{mathred}{o}}\color{mathred}{\:}{\color{mathred}{p}\color{mathred}{i}\color{mathred}{e}\color{mathred}{c}\color{mathred}{e}\color{mathred}{s}}. \\ $$

Commented bybenjo_mathlover last updated on 04/Mar/21

$$ \\ $$ hello sir, that's the problem written in the book no other explanation\\n

Commented bymr W last updated on 04/Mar/21

then it′s assumed that a box (here a  child) may be empty. in this case  there are C_4 ^(25+4) =((29!)/(25!4!))=23751 ways.

$${then}\:{it}'{s}\:{assumed}\:{that}\:{a}\:{box}\:\left({here}\:{a}\right. \\ $$ $$\left.{child}\right)\:{may}\:{be}\:{empty}.\:{in}\:{this}\:{case} \\ $$ $${there}\:{are}\:{C}_{\mathrm{4}} ^{\mathrm{25}+\mathrm{4}} =\frac{\mathrm{29}!}{\mathrm{25}!\mathrm{4}!}=\mathrm{23751}\:{ways}. \\ $$

Commented bymr W last updated on 04/Mar/21

if each child must get at least  two pieces, then we have in fact only  15 pieces to distribute, there are  C_4 ^(15+4) =3876 ways.

$${\color{mathred}{i}\color{mathred}{f}}\color{mathred}{\:}{\color{mathred}{e}\color{mathred}{a}\color{mathred}{c}\color{mathred}{h}}\color{mathred}{\:}{\color{mathred}{c}\color{mathred}{h}\color{mathred}{i}\color{mathred}{l}\color{mathred}{d}}\color{mathred}{\:}{\color{mathred}{m}\color{mathred}{u}\color{mathred}{s}\color{mathred}{t}}\color{mathred}{\:}{\color{mathred}{g}\color{mathred}{e}\color{mathred}{t}}\color{mathred}{\:}{\color{mathred}{a}\color{mathred}{t}}\color{mathred}{\:}{\color{mathred}{l}\color{mathred}{e}\color{mathred}{a}\color{mathred}{s}\color{mathred}{t}} \\ $$ $${\color{mathred}{t}\color{mathred}{w}\color{mathred}{o}}\color{mathred}{\:}{\color{mathred}{p}\color{mathred}{i}\color{mathred}{e}\color{mathred}{c}\color{mathred}{e}\color{mathred}{s}},\:{then}\:{we}\:{have}\:{in}\:{fact}\:{only} \\ $$ $$\mathrm{15}\:{pieces}\:{to}\:{distribute},\:{there}\:{are} \\ $$ $${C}_{\mathrm{4}} ^{\mathrm{15}+\mathrm{4}} =\mathrm{3876}\:{ways}. \\ $$

Commented bybenjo_mathlover last updated on 04/Mar/21

ok sir. thanks

$$\mathrm{ok}\:\mathrm{sir}.\:\mathrm{thanks} \\ $$

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