Question Number 195964 by mr W last updated on 15/Aug/23
$${the}\:{family}\:{A}\:{has}\:\mathrm{5}\:{members}\:{and}\:{the} \\ $$$${family}\:{B}\:{has}\:\mathrm{4}\:{members}.\:{there}\:{are}\: \\ $$$$\mathrm{6}\:{personsfrom}\:{other}\:{families}. \\ $$$${in}\:{how}\:{many}\:{ways}\:{can}\:{you}\:{arrange} \\ $$$${these}\:\mathrm{15}\:{persons}\:{around}\:{a}\:{round}\:{table} \\ $$$${such}\:{that}\:{no}\:{member}\:{from}\:{family}\:{A} \\ $$$${and}\:{no}\:{member}\:{from}\:{family}\:{B}\:{are} \\ $$$${next}\:{to}\:{each}\:{other}? \\ $$
Answered by mr W last updated on 15/Aug/23
Commented by mr W last updated on 15/Aug/23
$${we}\:{arrange}\:{at}\:{first}\:{the}\:\mathrm{6}\:{people}\:{from} \\ $$$${other}\:{families}\:\left({green}\:{seats}\right)\:{on} \\ $$$${the}\:{table}.\:{there}\:{are}\:\mathrm{5}!\:{ways}\:{to}\:{do}\:{this}. \\ $$$${say}\:\spadesuit\:{stands}\:{for}\:{a}\:{seat}\:{for}\:{a}\:{member} \\ $$$${from}\:{family}\:{A}\:{and}\:\clubsuit\:{stands}\:{for}\:{a} \\ $$$${seat}\:{for}\:{a}\:{member}\:{from}\:{family}\:{B}. \\ $$$${each}\:{of}\:{the}\:\mathrm{6}\:{places}\:\left({blue}\:{boxes}\right)\: \\ $$$${between}\:{the}\:{green}\:{seats}\:{can}\:{be}\:{filled} \\ $$$${with}: \\ $$$${nothing}\:\left({empty}\right)\:\Leftrightarrow\:{we}\:{add}\:\mathrm{0}\:{to}\:{it} \\ $$$${or}\:{a}\:{seat}\:\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{1}\:{to}\:{it} \\ $$$${or}\:\mathrm{2}\:{seats}\:\spadesuit\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{2}\:{to}\:{it} \\ $$$${or}\:\mathrm{3}\:{seats}\:\spadesuit\spadesuit\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{3}\:{to}\:{it} \\ $$$${or}\:\mathrm{4}\:{seats}\:\spadesuit\spadesuit\spadesuit\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{4}\:{to}\:{it} \\ $$$${or}\:\mathrm{5}\:{seats}\:\spadesuit\spadesuit\spadesuit\spadesuit\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{5}\:{to}\:{it} \\ $$$${or}\:{a}\:{seat}\:\clubsuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{100}\:{to}\:{it} \\ $$$${or}\:\mathrm{2}\:{seats}\:\clubsuit\clubsuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{200}\:{to}\:{it} \\ $$$${or}\:\mathrm{3}\:{seats}\:\clubsuit\clubsuit\clubsuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{300}\:{to}\:{it} \\ $$$${or}\:\mathrm{4}\:{seats}\:\clubsuit\clubsuit\clubsuit\clubsuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{400}\:{to}\:{it} \\ $$$${in}\:{all}\:\mathrm{6}\:{boxes}\:{we}\:{should}\:{totally}\:{put}\: \\ $$$$\mathrm{5}\:{seats}\:\spadesuit\spadesuit\spadesuit\spadesuit\spadesuit\:{and}\:\mathrm{4}\:{seats}\:\clubsuit\clubsuit\clubsuit\clubsuit. \\ $$$${that}\:{means}\:{the}\:{sum}\:{of}\:{all}\:\mathrm{6}\:{boxes} \\ $$$${should}\:{be}\:\mathrm{5}+\mathrm{400}=\mathrm{405}.\:{so}\:{the}\:{number} \\ $$$${of}\:{ways}\:{to}\:{do}\:{this}\:{is}\:{the}\:{coefficient} \\ $$$${of}\:{term}\:{x}^{\mathrm{405}} \:{in}\:{the}\:{expansion}\:{of}\:{the} \\ $$$${generating}\:{function} \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +{x}^{\mathrm{4}} +{x}^{\mathrm{5}} +{x}^{\mathrm{100}} +{x}^{\mathrm{200}} +{x}^{\mathrm{300}} +{x}^{\mathrm{400}} \right)^{\mathrm{6}} , \\ $$$${which}\:{is}\:\mathrm{4626},\:{see}\:{below}.\:{since}\:{the}\: \\ $$$${members}\:{both}\:{in}\:{family}\:{A}\:{and}\: \\ $$$${family}\:{B}\:{are}\:{permutable},\:{the}\:{number} \\ $$$${of}\:{ways}\:{to}\:{arrange}\:{the}\:{members}\: \\ $$$${from}\:{family}\:{A}\:{and}\:{B}\:{is}\:\mathrm{4626}×\mathrm{4}!×\mathrm{5}!. \\ $$$${the}\:{totaly}\:{number}\:{of}\:{ways}\:{to}\:{arrange} \\ $$$${all}\:{people}\:{around}\:{the}\:{table}\:{is}\:{thus} \\ $$$$\mathrm{4626}×\mathrm{4}!×\mathrm{5}!×\mathrm{5}!=\mathrm{1}\:\mathrm{598}\:\mathrm{745}\:\mathrm{600} \\ $$
Commented by mr W last updated on 15/Aug/23
Commented by MM42 last updated on 15/Aug/23
$${sir}\:\boldsymbol{{W}} \\ $$$${how}\:{to}\:{draw}\:{geometric}\:{shapes}\: \\ $$$${on}\:{this}\:{page}?{as}\:{in}\:{the}\:{figure}\:{above}\: \\ $$
Commented by mr W last updated on 15/Aug/23
$${try}\:{LEKH}\:{diagram}. \\ $$
Commented by MM42 last updated on 15/Aug/23
$${thank}\:{you}\:{so} \\ $$
Commented by Tawa11 last updated on 16/Aug/23
$$\mathrm{Great}\:\mathrm{work}\:\mathrm{sir}. \\ $$
Commented by York12 last updated on 16/Aug/23
$${sir}\:{where}\:{to}\:{learn}\:{combinatorics} \\ $$
Commented by York12 last updated on 16/Aug/23
$$ \\ $$$${or}\:{a}\:{seat}\:\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{1}\:{to}\:{it} \\ $$$${or}\:\mathrm{2}\:{seats}\:\spadesuit\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{2}\:{to}\:{it} \\ $$$${or}\:\mathrm{3}\:{seats}\:\spadesuit\spadesuit\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{3}\:{to}\:{it} \\ $$$${or}\:\mathrm{4}\:{seats}\:\spadesuit\spadesuit\spadesuit\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{4}\:{to}\:{it} \\ $$$${or}\:\mathrm{5}\:{seats}\:\spadesuit\spadesuit\spadesuit\spadesuit\spadesuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{5}\:{to}\:{it} \\ $$$${or}\:{a}\:{seat}\:\clubsuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{100}\:{to}\:{it} \\ $$$${or}\:\mathrm{2}\:{seats}\:\clubsuit\clubsuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{200}\:{to}\:{it} \\ $$$${or}\:\mathrm{3}\:{seats}\:\clubsuit\clubsuit\clubsuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{300}\:{to}\:{it} \\ $$$${or}\:\mathrm{4}\:{seats}\:\clubsuit\clubsuit\clubsuit\clubsuit\:\Leftrightarrow\:{we}\:{add}\:\mathrm{400}\:{to}\:{it} \\ $$$${in}\:{all}\:\mathrm{6}\:{boxes}\:{we}\:{should}\:{totally}\:{put}\: \\ $$$$\mathrm{5}\:{seats}\:\spadesuit\spadesuit\spadesuit\spadesuit\spadesuit\:{and}\:\mathrm{4}\:{seats}\:\clubsuit\clubsuit\clubsuit\clubsuit. \\ $$$${that}\:{means}\:{the}\:{sum}\:{of}\:{all}\:\mathrm{6}\:{boxes} \\ $$$${should}\:{be}\:\mathrm{5}+\mathrm{400}=\mathrm{405} \\ $$$$ \\ $$
Commented by York12 last updated on 16/Aug/23
$${I}\:{do}\:{not}\:{understand}\:{what}\:{do}\:{you} \\ $$$${mean}\:{by}\:{this}\:{part} \\ $$$$ \\ $$
Commented by mr W last updated on 16/Aug/23
$${you}\:{have}\:\mathrm{5}\:{seats}\:\spadesuit{for}\:{the}\:{family}\:{A}. \\ $$$${in}\:{each}\:{box}\:{you}\:{can}\:{put}\:\mathrm{1}\:{or}\:\mathrm{2}\:{or}\:\mathrm{3}\:{or} \\ $$$$\mathrm{4}\:{or}\:\mathrm{5}\:{seats}\:\spadesuit.\:{totally}\:{you}\:{must}\:{put} \\ $$$${all}\:\:\mathrm{5}\:{seats}\:\spadesuit.\:{or}\:{you}\:{put}\:{in}\:{this} \\ $$$${only}\:{seats}\:{for}\:{family}\:{B},\:{namly} \\ $$$$\mathrm{1}\:{or}\:\mathrm{2}\:{or}\:\mathrm{3}\:{or}\:\mathrm{4}\:{seats}\:\clubsuit.\:{totally}\:{you} \\ $$$${must}\:{put}\:{all}\:\mathrm{4}\:{seats}\:\clubsuit. \\ $$$${if}\:{empty}\:{means}\:\mathrm{0},\:{a}\:\spadesuit\:{means}\:\mathrm{1}, \\ $$$$\mathrm{2}\:\spadesuit\spadesuit\:{means}\:\mathrm{2},\:\mathrm{3}\:\spadesuit\spadesuit\spadesuit\:{means}\:\mathrm{3},\: \\ $$$$\mathrm{4}\:\spadesuit\spadesuit\spadesuit\spadesuit\:{means}\:\mathrm{4},\:\mathrm{5}\:\spadesuit\spadesuit\spadesuit\spadesuit\spadesuit\:{means} \\ $$$$\mathrm{5},\:{and}\:{a}\:\clubsuit\:{means}\:\mathrm{100},\:\mathrm{2}\:\clubsuit\clubsuit\:{means} \\ $$$$\mathrm{200},\:\mathrm{3}\:\clubsuit\clubsuit\clubsuit\:{means}\:\mathrm{300},\:\mathrm{4}\:\clubsuit\clubsuit\clubsuit\clubsuit\: \\ $$$${means}\:\mathrm{400},\:{the}\:{sum}\:{of}\:{numbers}\:{in} \\ $$$${all}\:\mathrm{6}\:{boxes}\:{must}\:{be}\:\mathrm{405}. \\ $$
Commented by York12 last updated on 16/Aug/23
$${thanks} \\ $$$$ \\ $$