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n married couples are invited to a dance party. for the first dance n paires are radomly selected. what′s the probability that no woman dances with her own husband? 1) if a pair must be of different genders. 2) if a pair can also be of the same gender.

$$\boldsymbol{{n}}\:{married}\:{couples}\:{are}\:{invited}\:{to} \\ $$$${a}\:{dance}\:{party}.\:{for}\:{the}\:{first}\:{dance} \\ $$$$\boldsymbol{{n}}\:{paires}\:{are}\:{radomly}\:{selected}.\: \\ $$$${what}'{s}\:{the}\:{probability}\:{that}\:{no}\:{woman} \\ $$$${dances}\:{with}\:{her}\:{own}\:{husband}? \\ $$$$\left.\mathrm{1}\right)\:{if}\:{a}\:{pair}\:{must}\:{be}\:{of}\:{different} \\ $$$$\:\:\:\:\:{genders}. \\ $$$$\left.\mathrm{2}\right)\:{if}\:{a}\:{pair}\:{can}\:{also}\:{be}\:{of}\:{the}\:{same}\: \\ $$$$\:\:\:\:\:{gender}. \\ $$

Question Number 207779 Answers: 1 Comments: 0

Question Number 207774 Answers: 1 Comments: 4

Question Number 207771 Answers: 1 Comments: 0

Simplify: (((b)^(1/4) (√c) − (c)^(1/4) (√b))/( (b)^(1/4) (√c) − (c)^(1/4) )) = ?

$$\mathrm{Simplify}:\:\:\:\:\:\frac{\sqrt[{\mathrm{4}}]{\boldsymbol{\mathrm{b}}}\:\:\sqrt{\boldsymbol{\mathrm{c}}}\:\:−\:\:\sqrt[{\mathrm{4}}]{\boldsymbol{\mathrm{c}}}\:\:\sqrt{\boldsymbol{\mathrm{b}}}}{\:\sqrt[{\mathrm{4}}]{\boldsymbol{\mathrm{b}}}\:\:\sqrt{\boldsymbol{\mathrm{c}}}\:\:−\:\:\sqrt[{\mathrm{4}}]{\boldsymbol{\mathrm{c}}}}\:\:=\:\:? \\ $$

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Question Number 207753 Answers: 1 Comments: 0

Question Number 207752 Answers: 1 Comments: 3

Two ships have the same berth in a port. It is known that the arrival times of the two ships are independent and have the same probability of docking on a Sunday (00.00−24.00) If the berth time of the first ship is 2 hours and the berth time of the second ship is 4 hours, the probability that one ship will have to wait until the berth can be used is □ ((67)/(144)) □ ((67)/(288)) □ (1/4) □((33)/(144))

$$\:\mathrm{Two}\:\mathrm{ships}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{berth}\: \\ $$$$\:\mathrm{in}\:\mathrm{a}\:\mathrm{port}.\:\mathrm{It}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\:\mathrm{arrival}\:\mathrm{times}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two}\:\mathrm{ships}\: \\ $$$$\:\mathrm{are}\:\mathrm{independent}\:\mathrm{and}\:\mathrm{have}\:\mathrm{the}\: \\ $$$$\:\mathrm{same}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{docking}\: \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{Sunday}\:\left(\mathrm{00}.\mathrm{00}−\mathrm{24}.\mathrm{00}\right) \\ $$$$\:\mathrm{If}\:\mathrm{the}\:\mathrm{berth}\:\mathrm{time}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{ship} \\ $$$$\:\mathrm{is}\:\mathrm{2}\:\mathrm{hours}\:\mathrm{and}\:\mathrm{the}\:\mathrm{berth}\:\mathrm{time} \\ $$$$\:\mathrm{of}\:\mathrm{the}\:\mathrm{second}\:\mathrm{ship}\:\mathrm{is}\:\mathrm{4}\:\mathrm{hours},\: \\ $$$$\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{one}\:\mathrm{ship} \\ $$$$\:\mathrm{will}\:\mathrm{have}\:\mathrm{to}\:\mathrm{wait}\:\mathrm{until}\:\mathrm{the} \\ $$$$\:\mathrm{berth}\:\mathrm{can}\:\mathrm{be}\:\mathrm{used}\:\mathrm{is}\: \\ $$$$\:\Box\:\frac{\mathrm{67}}{\mathrm{144}}\:\:\:\:\:\Box\:\frac{\mathrm{67}}{\mathrm{288}}\:\:\:\:\Box\:\frac{\mathrm{1}}{\mathrm{4}}\:\:\:\:\Box\frac{\mathrm{33}}{\mathrm{144}} \\ $$

Question Number 207751 Answers: 0 Comments: 0

It is known that a balanced 6−sided dice originally had 2,3,4,5,6 and 7. The dice wre thrown once and the result was observed. If an odd numbers appears, than the number is replaced with the number 8. However, if an even number appears , the number is replaced with the number 1. Then the dice whose dice have been replaced are thrown again, the probability of an odd dice odd dice appearing is □ (1/3) □ (2/3) □ (1/2) □ 1

$$\:\:\mathrm{It}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{a}\:\mathrm{balanced}\:\mathrm{6}−\mathrm{sided}\: \\ $$$$\:\mathrm{dice}\:\mathrm{originally}\:\mathrm{had}\:\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}\:\mathrm{and}\:\mathrm{7}. \\ $$$$\:\mathrm{The}\:\mathrm{dice}\:\mathrm{wre}\:\mathrm{thrown}\:\mathrm{once}\:\mathrm{and}\: \\ $$$$\:\mathrm{the}\:\mathrm{result}\:\mathrm{was}\:\mathrm{observed}.\:\mathrm{If}\:\mathrm{an}\: \\ $$$$\mathrm{odd}\:\mathrm{numbers}\:\mathrm{appears},\:\mathrm{than}\:\mathrm{the}\: \\ $$$$\:\mathrm{number}\:\mathrm{is}\:\mathrm{replaced}\:\mathrm{with}\:\mathrm{the}\: \\ $$$$\:\mathrm{number}\:\mathrm{8}.\:\mathrm{However},\:\mathrm{if}\:\mathrm{an}\:\mathrm{even}\: \\ $$$$\:\mathrm{number}\:\mathrm{appears}\:,\:\mathrm{the}\:\mathrm{number} \\ $$$$\:\mathrm{is}\:\mathrm{replaced}\:\mathrm{with}\:\mathrm{the}\:\mathrm{number}\:\mathrm{1}. \\ $$$$\:\mathrm{Then}\:\mathrm{the}\:\mathrm{dice}\:\mathrm{whose}\:\mathrm{dice}\:\mathrm{have}\: \\ $$$$\:\mathrm{been}\:\mathrm{replaced}\:\mathrm{are}\:\mathrm{thrown}\:\mathrm{again},\: \\ $$$$\:\mathrm{the}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{dice}\: \\ $$$$\:\mathrm{odd}\:\mathrm{dice}\:\mathrm{appearing}\:\mathrm{is}\: \\ $$$$\:\:\Box\:\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\Box\:\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\Box\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\Box\:\mathrm{1}\: \\ $$

Question Number 207747 Answers: 3 Comments: 0

find the value of y given that y^(y ) =3^(y+9)

$${find}\:{the}\:{value}\:{of}\:{y}\:{given}\:{that}\:{y}^{{y}\:} =\mathrm{3}^{{y}+\mathrm{9}} \\ $$

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X, Y and Z are points on the sides AB, BC and AC of the triangle ABC, such that AX:XB =4:3, BY:YC=2:3, CZ:ZA=2:1. Find the ratio of the area of the triangle XYZ to that of triangle ABC.

$${X},\:{Y}\:{and}\:{Z}\:{are}\:{points}\:{on}\:{the}\:{sides}\:{AB}, \\ $$$${BC}\:{and}\:{AC}\:{of}\:{the}\:{triangle}\:{ABC},\:{such} \\ $$$${that}\:{AX}:{XB}\:=\mathrm{4}:\mathrm{3},\:{BY}:{YC}=\mathrm{2}:\mathrm{3},\: \\ $$$${CZ}:{ZA}=\mathrm{2}:\mathrm{1}.\:{Find}\:{the}\:{ratio}\:{of}\:{the}\:{area} \\ $$$${of}\:{the}\:{triangle}\:{XYZ}\:{to}\:{that}\:{of}\:{triangle} \\ $$$${ABC}. \\ $$

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