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Question Number 55344 by Necxx last updated on 21/Feb/19
A whatsapp group contains 7 women  and 3 men.If they are leaving  the group one at a time from the  group what is the probability of a  woman leaving then a man leaving  and so on alternately until only a  woman is remaining?
$${A}\:{whatsapp}\:{group}\:{contains}\:\mathrm{7}\:{women} \\ $$$${and}\:\mathrm{3}\:{men}.{If}\:{they}\:{are}\:{leaving} \\ $$$${the}\:{group}\:{one}\:{at}\:{a}\:{time}\:{from}\:{the} \\ $$$${group}\:{what}\:{is}\:{the}\:{probability}\:{of}\:{a} \\ $$$${woman}\:{leaving}\:{then}\:{a}\:{man}\:{leaving} \\ $$$${and}\:{so}\:{on}\:{alternately}\:{until}\:{only}\:{a} \\ $$$${woman}\:{is}\:{remaining}? \\ $$$$ \\ $$
Commented by mr W last updated on 22/Feb/19
WMWMWMWWWW → leaving  ((7×3×6×2×5×1×4!)/(10!))  =((3×2)/(10×9×8))  =(1/(120))
$${WMWMWMWWWW}\:\rightarrow\:{leaving} \\ $$$$\frac{\mathrm{7}×\mathrm{3}×\mathrm{6}×\mathrm{2}×\mathrm{5}×\mathrm{1}×\mathrm{4}!}{\mathrm{10}!} \\ $$$$=\frac{\mathrm{3}×\mathrm{2}}{\mathrm{10}×\mathrm{9}×\mathrm{8}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{120}} \\ $$
Commented by necx1 last updated on 23/Feb/19
yeah.... Thanks
$${yeah}….\:{Thanks} \\ $$
Commented by rahul 19 last updated on 23/Feb/19
Sir, pl explain more what  you have done.
$${Sir},\:{pl}\:{explain}\:{more}\:{what}\:\:{you}\:{have}\:{done}. \\ $$
Commented by mr W last updated on 24/Feb/19
total number of ways to arrange 10  persons is 10!.  such that the persons leave whatsapp  in the way as requested in the question,  the persons should be arranged like  this:  WMWMWMWWWW→leaving whatsapp  with W=woman, M=man  there are 7!3! such arrangements,  therefore the probability is  ((7!3!)/(10!))=((3!)/(10×9×8))=(1/(120))
$${total}\:{number}\:{of}\:{ways}\:{to}\:{arrange}\:\mathrm{10} \\ $$$${persons}\:{is}\:\mathrm{10}!. \\ $$$${such}\:{that}\:{the}\:{persons}\:{leave}\:{whatsapp} \\ $$$${in}\:{the}\:{way}\:{as}\:{requested}\:{in}\:{the}\:{question}, \\ $$$${the}\:{persons}\:{should}\:{be}\:{arranged}\:{like} \\ $$$${this}: \\ $$$${WMWMWMWWWW}\rightarrow{leaving}\:{whatsapp} \\ $$$${with}\:{W}={woman},\:{M}={man} \\ $$$${there}\:{are}\:\mathrm{7}!\mathrm{3}!\:{such}\:{arrangements}, \\ $$$${therefore}\:{the}\:{probability}\:{is} \\ $$$$\frac{\mathrm{7}!\mathrm{3}!}{\mathrm{10}!}=\frac{\mathrm{3}!}{\mathrm{10}×\mathrm{9}×\mathrm{8}}=\frac{\mathrm{1}}{\mathrm{120}} \\ $$
Commented by rahul 19 last updated on 25/Feb/19
thank U sir!
$${thank}\:{U}\:{sir}! \\ $$
Commented by rahul 19 last updated on 25/Feb/19
Today is 25th Feb'19 ... why my comment shows 24th Feb'19 ? some technical error ...

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