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Question Number 149955 by jlewis last updated on 08/Aug/21

If a group consist of 8 men and 6 women, in how many ways can a committee of 5 be selected if: i) the committee is to consist of 3 men and 3 women. ii) there are no restrictions on the number of men and women on the committee. iii) at least one man

$$\mathrm{If}\:\mathrm{a}\:\mathrm{group}\:\mathrm{consist}\:\mathrm{of}\:\mathrm{8}\:\mathrm{men}\:\mathrm{and}\:\mathrm{6}\:\mathrm{women}, \\ $$$$\mathrm{in}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{a}\:\mathrm{committee}\:\mathrm{of} \\ $$$$\mathrm{5}\:\mathrm{be}\:\mathrm{selected}\:\mathrm{if}: \\ $$$$\left.\:\:\:\:\:\:\mathrm{i}\right)\:\mathrm{the}\:\mathrm{committee}\:\mathrm{is}\:\mathrm{to}\:\mathrm{consist}\:\mathrm{of}\:\mathrm{3}\:\mathrm{men} \\ $$$$\mathrm{and}\:\mathrm{3}\:\mathrm{women}. \\ $$$$\left.\:\:\:\:\:\:\mathrm{ii}\right)\:\mathrm{there}\:\mathrm{are}\:\mathrm{no}\:\mathrm{restrictions}\:\mathrm{on}\:\mathrm{the}\: \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{men}\:\mathrm{and}\:\mathrm{women}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{committee}. \\ $$$$\left.\:\:\:\:\:\:\:\mathrm{iii}\right)\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{man} \\ $$

Answered by mr W last updated on 08/Aug/21

i) C_3 ^8 ×C_3 ^6 =1120 ways ii) C_6 ^(14) =3003 ways iii) C_6 ^(14) −C_6 ^8 −C_1 ^6 C_5 ^8 =2639 ways

$$\left.{i}\right) \\ $$$${C}_{\mathrm{3}} ^{\mathrm{8}} ×{C}_{\mathrm{3}} ^{\mathrm{6}} =\mathrm{1120}\:{ways} \\ $$$$\left.{ii}\right) \\ $$$${C}_{\mathrm{6}} ^{\mathrm{14}} =\mathrm{3003}\:{ways} \\ $$$$\left.{iii}\right) \\ $$$${C}_{\mathrm{6}} ^{\mathrm{14}} −{C}_{\mathrm{6}} ^{\mathrm{8}} −{C}_{\mathrm{1}} ^{\mathrm{6}} {C}_{\mathrm{5}} ^{\mathrm{8}} =\mathrm{2639}\:{ways} \\ $$

Commented by jlewis last updated on 08/Aug/21

hello sir,i gauze it was a commettee of 5 not 6

$$\mathrm{hello}\:\mathrm{sir},\mathrm{i}\:\mathrm{gauze}\:\mathrm{it}\:\mathrm{was}\:\mathrm{a}\:\mathrm{commettee} \\ $$$$\mathrm{of}\:\mathrm{5}\:\mathrm{not}\:\mathrm{6} \\ $$

Commented by jlewis last updated on 08/Aug/21

and also if u are asked.at least one of each sex

$$\mathrm{and}\:\mathrm{also}\:\mathrm{if}\:\mathrm{u}\:\mathrm{are}\:\mathrm{asked}.\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{of} \\ $$$$\:\mathrm{each}\:\mathrm{sex} \\ $$

Commented by mr W last updated on 09/Aug/21

you said i) the committee is to consist of 3 men and 3 women. so i assumed that the committee consists of 6 people.

$${you}\:{said} \\ $$$$\left.\mathrm{i}\right)\:\mathrm{the}\:\mathrm{committee}\:\mathrm{is}\:\mathrm{to}\:\mathrm{consist}\:\mathrm{of}\:\mathrm{3}\:\mathrm{men} \\ $$$$\mathrm{and}\:\mathrm{3}\:\mathrm{women}. \\ $$$${so}\:{i}\:{assumed}\:{that}\:{the}\:{committee} \\ $$$${consists}\:{of}\:\mathrm{6}\:{people}. \\ $$

Commented by mr W last updated on 09/Aug/21

you said iii) at least one man which doesn′t mean at least one of each sex.

$${you}\:{said} \\ $$$$\left.\mathrm{iii}\right)\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{man} \\ $$$${which}\:{doesn}'{t}\:{mean}\:{at}\:{least}\:{one}\:{of} \\ $$$${each}\:{sex}. \\ $$

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