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Question Number 115230 by bemath last updated on 24/Sep/20

2 women and 4 men will sit on the  8 available seats and surround   the round table . The many possible  arrangements of them sitting  if they sat randomly

$$\mathrm{2}\:{women}\:{and}\:\mathrm{4}\:{men}\:{will}\:{sit}\:{on}\:{the} \\ $$$$\mathrm{8}\:{available}\:{seats}\:{and}\:{surround}\: \\ $$$${the}\:{round}\:{table}\:.\:{The}\:{many}\:{possible} \\ $$$${arrangements}\:{of}\:{them}\:{sitting} \\ $$$${if}\:{they}\:{sat}\:{randomly} \\ $$

Commented by mr W last updated on 24/Sep/20

P_5 ^7 =2520  or C_5 ^7 ×5!

$${P}_{\mathrm{5}} ^{\mathrm{7}} =\mathrm{2520} \\ $$$${or}\:{C}_{\mathrm{5}} ^{\mathrm{7}} ×\mathrm{5}! \\ $$

Commented by bemath last updated on 24/Sep/20

sir. what if the two woman always  sat side by side?

$${sir}.\:{what}\:{if}\:{the}\:{two}\:{woman}\:{always} \\ $$$${sat}\:{side}\:{by}\:{side}? \\ $$

Commented by bemath last updated on 24/Sep/20

=^8 C_6  × 5! = ((8!)/(6!.2!)) ×5! = 28×120   = 3360. why different sir?

$$=^{\mathrm{8}} {C}_{\mathrm{6}} \:×\:\mathrm{5}!\:=\:\frac{\mathrm{8}!}{\mathrm{6}!.\mathrm{2}!}\:×\mathrm{5}!\:=\:\mathrm{28}×\mathrm{120} \\ $$$$\:=\:\mathrm{3360}.\:{why}\:{different}\:{sir}? \\ $$

Commented by mr W last updated on 24/Sep/20

2×P_4 ^6 =720

$$\mathrm{2}×{P}_{\mathrm{4}} ^{\mathrm{6}} =\mathrm{720} \\ $$

Commented by mr W last updated on 24/Sep/20

to select 6 seats from 8 seats on a  round table there are C_5 ^7  ways, not  C_6 ^8  ways! because the “absolute”   positions of the seats are not important,  but only the “relative” positions of  them. that means you can fix one  seat and select 5 from the remaining  7 seats, you have C_5 ^7  ways to do this.

$${to}\:{select}\:\mathrm{6}\:{seats}\:{from}\:\mathrm{8}\:{seats}\:{on}\:{a} \\ $$$${round}\:{table}\:{there}\:{are}\:{C}_{\mathrm{5}} ^{\mathrm{7}} \:{ways},\:{not} \\ $$$${C}_{\mathrm{6}} ^{\mathrm{8}} \:{ways}!\:{because}\:{the}\:``{absolute}''\: \\ $$$${positions}\:{of}\:{the}\:{seats}\:{are}\:{not}\:{important}, \\ $$$${but}\:{only}\:{the}\:``{relative}''\:{positions}\:{of} \\ $$$${them}.\:{that}\:{means}\:{you}\:{can}\:{fix}\:{one} \\ $$$${seat}\:{and}\:{select}\:\mathrm{5}\:{from}\:{the}\:{remaining} \\ $$$$\mathrm{7}\:{seats},\:{you}\:{have}\:{C}_{\mathrm{5}} ^{\mathrm{7}} \:{ways}\:{to}\:{do}\:{this}. \\ $$

Commented by bemath last updated on 25/Sep/20

thank you prof for explanation

$${thank}\:{you}\:{prof}\:{for}\:{explanation} \\ $$

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