3-men-and-4-women-are-to-sit-on-a-table-Calculate-the-number-of-possible-sitting-arrangements-if-a-they-sit-in-a-row-such-that-the-men-must-not-sit-next-to-each-other-b-they-

Question Number 167979 by nadovic last updated on 31/Mar/22

$$\mathrm{3}\:\mathrm{men}\:\mathrm{and}\:\mathrm{4}\:\mathrm{women}\:\mathrm{are}\:\mathrm{to}\:\mathrm{sit}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{table}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{possible}\:\mathrm{sitting}\:\mathrm{arrangements}\:\mathrm{if} \\ $$$$\:\left({a}\right)\:\mathrm{they}\:\mathrm{sit}\:\mathrm{in}\:\mathrm{a}\:\mathrm{row}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{men}\:\mathrm{must}\:\mathrm{not}\:\mathrm{sit}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{other}. \\ $$$$\:\left({b}\right)\:\mathrm{they}\:\mathrm{sit}\:\mathrm{in}\:\mathrm{circular}\:\mathrm{pattern}\:\mathrm{and} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{the}\:\mathrm{clockwise}\:\mathrm{and}\:\mathrm{anticlockwise} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{orders}\:\mathrm{are}\:\mathrm{considered}\:\mathrm{the}\:\mathrm{same}. \\ $$

Answered by mr W last updated on 31/Mar/22

(a) □W□W□W□W□ C_5 ^3 ×3!×4!=1440 arrangements

$$\left({a}\right) \\ $$$$\square{W}\square{W}\square{W}\square{W}\square \\ $$$${C}_{\mathrm{5}} ^{\mathrm{3}} ×\mathrm{3}!×\mathrm{4}!=\mathrm{1440}\:{arrangements} \\ $$

Commented by mr W last updated on 31/Mar/22