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Question Number 135218 by benjo_mathlover last updated on 11/Mar/21
$$\mathrm{Permutation} \\ $$How many ways can 10 men and 7 women sit at a round table so that no 2 women are next to each other? 😎😎😎😎
Answered by EDWIN88 last updated on 11/Mar/21
$$\Leftrightarrow\: \::\:\mathrm{place}\:\mathrm{of}\:\mathrm{woman},\:\mathrm{C}_{\mathrm{7}} ^{\:\mathrm{10}} \:=\:\mathrm{120} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{arrangement}\: \\ $$$$\mathrm{is}\:=\:\mathrm{7}!×\mathrm{9}!×\mathrm{120}\: \\ $$
Commented by EDWIN88 last updated on 11/Mar/21
Commented by EDWIN88 last updated on 11/Mar/21
$$\mathrm{typo} \\ $$
Answered by mr W last updated on 11/Mar/21
$${we}\:{place}\:{at}\:{first}\:{the}\:\mathrm{10}\:{men}.\:{there} \\ $$$${are}\:\mathrm{9}!\:{ways}.\:{then}\:{we}\:{place}\:{the}\:\mathrm{7} \\ $$$${women}\:{in}\:{the}\:\mathrm{10}\:{positions}\:{among} \\ $$$${the}\:{men},\:{there}\:{are}\:{C}_{\mathrm{7}} ^{\mathrm{10}} ×\mathrm{7}!\:{ways}. \\ $$$${totally}\:\mathrm{9}!×\mathrm{7}!×{C}_{\mathrm{7}} ^{\mathrm{10}} \\ $$