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Category: Permutation and Combination

In-a-group-of-ten-persons-each-person-is-asked-to-write-the-sum-of-the-ages-of-all-the-other-9-persons-If-all-the-ten-sums-form-the-9-element-set-82-83-84-85-87-90-91-92-find-the-individual

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Question Number 21685 by Tinkutara last updated on 30/Sep/17 $$\mathrm{In}\:\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\mathrm{ten}\:\mathrm{persons},\:\mathrm{each}\:\mathrm{person} \\ $$$$\mathrm{is}\:\mathrm{asked}\:\mathrm{to}\:\mathrm{write}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ages}\:\mathrm{of} \\ $$$$\mathrm{all}\:\mathrm{the}\:\mathrm{other}\:\mathrm{9}\:\mathrm{persons}.\:\mathrm{If}\:\mathrm{all}\:\mathrm{the}\:\mathrm{ten} \\ $$$$\mathrm{sums}\:\mathrm{form}\:\mathrm{the}\:\mathrm{9}-\mathrm{element}\:\mathrm{set}\:\left\{\mathrm{82},\:\mathrm{83},\:\mathrm{84},\right. \\ $$$$\left.\mathrm{85},\:\mathrm{87},\:\mathrm{90},\:\mathrm{91},\:\mathrm{92}\right\}\:\mathrm{find}\:\mathrm{the}\:\mathrm{individual} \\ $$$$\mathrm{ages}\:\mathrm{of}\:\mathrm{the}\:\mathrm{persons}\:\left(\mathrm{assuming}\:\mathrm{them}\:\mathrm{to}\right. \\ $$$$\left.\mathrm{be}\:\mathrm{whole}\:\mathrm{numbers}\:\mathrm{of}\:\mathrm{years}\right). \\ $$ Terms…

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Suppose-A-1-A-2-A-20-is-a-20-sided-regular-polygon-How-many-non-isosceles-scalene-triangles-can-be-formed-whose-vertices-are-among-the-vertices-of-the-polygon-but-whose-sides-are-not-the-side

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Question Number 21683 by Tinkutara last updated on 30/Sep/17 $$\mathrm{Suppose}\:{A}_{\mathrm{1}} {A}_{\mathrm{2}} …{A}_{\mathrm{20}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{20}-\mathrm{sided}\:\mathrm{regular} \\ $$$$\mathrm{polygon}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{non}-\mathrm{isosceles} \\ $$$$\left(\mathrm{scalene}\right)\:\mathrm{triangles}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{whose} \\ $$$$\mathrm{vertices}\:\mathrm{are}\:\mathrm{among}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{polygon}\:\mathrm{but}\:\mathrm{whose}\:\mathrm{sides}\:\mathrm{are}\:\mathrm{not}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polygon}? \\ $$…

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if-in-the-expansion-of-1-x-n-the-coefficient-of-x-9-is-the-aritmetic-mean-of-the-coeficients-of-x-8-and-x-10-find-the-possible-value-of-n-where-n-is-a-positive-integer-

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Question Number 87175 by john santu last updated on 03/Apr/20 $$\mathrm{if}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{n}} \:\mathrm{the} \\ $$$$\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{9}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{aritmetic}\: \\ $$$$\mathrm{mean}\:\mathrm{of}\:\mathrm{the}\:\mathrm{coeficients}\:\mathrm{of}\:\mathrm{x}^{\mathrm{8}} \:\mathrm{and}\: \\ $$$$\mathrm{x}^{\mathrm{10}} .\:\mathrm{find}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{value}\: \\ $$$$\mathrm{of}\:\mathrm{n}\:\mathrm{where}\:\mathrm{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer} \\ $$…

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If-n-objects-are-arranged-in-a-row-then-find-the-number-of-ways-of-selecting-three-of-these-objects-so-that-no-two-of-them-are-next-to-each-other-

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Question Number 21587 by Tinkutara last updated on 28/Sep/17 $$\mathrm{If}\:{n}\:\mathrm{objects}\:\mathrm{are}\:\mathrm{arranged}\:\mathrm{in}\:\mathrm{a}\:\mathrm{row},\:\mathrm{then} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{selecting} \\ $$$$\mathrm{three}\:\mathrm{of}\:\mathrm{these}\:\mathrm{objects}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{two}\:\mathrm{of} \\ $$$$\mathrm{them}\:\mathrm{are}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}. \\ $$ Commented by mrW1 last updated on 29/Sep/17…

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what-is-coefficient-of-t-3-in-the-expanssion-1-t-6-1-t-3-

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Question Number 87074 by jagoll last updated on 02/Apr/20 $$\mathrm{what}\:\mathrm{is}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{t}^{\mathrm{3}} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{expanssion}\:\left\{\frac{\mathrm{1}−\mathrm{t}^{\mathrm{6}} }{\mathrm{1}−\mathrm{t}}\right\}^{\mathrm{3}} \: \\ $$ Commented by mr W last updated on 02/Apr/20 $${t}^{\mathrm{2}}…

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Four-dice-are-rolled-The-number-of-ways-in-which-at-least-one-die-shows-3-is-

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Question Number 21405 by Tinkutara last updated on 23/Sep/17 $$\mathrm{Four}\:\mathrm{dice}\:\mathrm{are}\:\mathrm{rolled}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{die}\:\mathrm{shows}\:\mathrm{3},\:\mathrm{is} \\ $$ Answered by myintkhaing last updated on 23/Sep/17 $$\mathrm{1}−\left(\frac{\mathrm{5}}{\mathrm{6}}×\frac{\mathrm{5}}{\mathrm{6}}×\frac{\mathrm{5}}{\mathrm{6}}×\frac{\mathrm{5}}{\mathrm{6}}\right)=\frac{\mathrm{671}}{\mathrm{1296}} \\ $$ Commented…

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