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Category: Permutation and Combination

The-result-of-11-chess-matches-as-win-lose-or-draw-are-to-be-forecast-Out-of-all-possible-forecasts-the-number-of-ways-in-which-8-correct-and-3-incorrect-results-can-be-forecast-is-

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Question Number 21771 by Tinkutara last updated on 03/Oct/17 $$\mathrm{The}\:\mathrm{result}\:\mathrm{of}\:\mathrm{11}\:\mathrm{chess}\:\mathrm{matches}\:\left(\mathrm{as}\:\mathrm{win},\right. \\ $$$$\left.\mathrm{lose}\:\mathrm{or}\:\mathrm{draw}\right)\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{forecast}.\:\mathrm{Out}\:\mathrm{of} \\ $$$$\mathrm{all}\:\mathrm{possible}\:\mathrm{forecasts},\:\mathrm{the}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:\mathrm{8}\:\mathrm{correct}\:\mathrm{and}\:\mathrm{3}\:\mathrm{incorrect} \\ $$$$\mathrm{results}\:\mathrm{can}\:\mathrm{be}\:\mathrm{forecast}\:\mathrm{is} \\ $$ Terms of Service Privacy Policy…

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There-are-3-apartments-A-B-and-C-for-rent-in-a-building-Each-apartment-will-accept-either-3-or-4-occupants-The-number-of-ways-of-renting-the-apartments-to-10-students-

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Question Number 21766 by Tinkutara last updated on 03/Oct/17 $$\mathrm{There}\:\mathrm{are}\:\mathrm{3}\:\mathrm{apartments}\:{A},\:{B}\:\mathrm{and}\:{C}\:\mathrm{for} \\ $$$$\mathrm{rent}\:\mathrm{in}\:\mathrm{a}\:\mathrm{building}.\:\mathrm{Each}\:\mathrm{apartment}\:\mathrm{will} \\ $$$$\mathrm{accept}\:\mathrm{either}\:\mathrm{3}\:\mathrm{or}\:\mathrm{4}\:\mathrm{occupants}.\:\mathrm{The} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{renting}\:\mathrm{the} \\ $$$$\mathrm{apartments}\:\mathrm{to}\:\mathrm{10}\:\mathrm{students} \\ $$ Answered by mrW1 last updated…

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Six-cards-and-six-envelopes-are-numbered-1-2-3-4-5-6-and-cards-are-to-be-placed-in-envelopes-so-that-each-envelope-contains-exactly-one-card-and-no-card-is-placed-in-the-envelope-bearing-the-same

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Question Number 21768 by Tinkutara last updated on 03/Oct/17 $$\mathrm{Six}\:\mathrm{cards}\:\mathrm{and}\:\mathrm{six}\:\mathrm{envelopes}\:\mathrm{are}\:\mathrm{numbered} \\ $$$$\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4},\:\mathrm{5},\:\mathrm{6}\:\mathrm{and}\:\mathrm{cards}\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{placed} \\ $$$$\mathrm{in}\:\mathrm{envelopes}\:\mathrm{so}\:\mathrm{that}\:\mathrm{each}\:\mathrm{envelope} \\ $$$$\mathrm{contains}\:\mathrm{exactly}\:\mathrm{one}\:\mathrm{card}\:\mathrm{and}\:\mathrm{no}\:\mathrm{card} \\ $$$$\mathrm{is}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{envelope}\:\mathrm{bearing}\:\mathrm{the} \\ $$$$\mathrm{same}\:\mathrm{number}\:\mathrm{and}\:\mathrm{moreover}\:\mathrm{the}\:\mathrm{card} \\ $$$$\mathrm{numbered}\:\mathrm{1}\:\mathrm{is}\:\mathrm{always}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{envelope} \\ $$$$\mathrm{numbered}\:\mathrm{2}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways} \\…

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There-are-m-points-on-the-line-AB-and-n-points-on-the-line-AC-excluding-the-point-A-Triangles-are-formed-joining-these-points-i-When-point-A-is-not-included-ii-When-point-A-is-included-The-rat

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Question Number 21760 by Tinkutara last updated on 03/Oct/17 $$\mathrm{There}\:\mathrm{are}\:{m}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}\:{AB}\:\mathrm{and} \\ $$$${n}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}\:{AC},\:\mathrm{excluding}\:\mathrm{the} \\ $$$$\mathrm{point}\:{A}.\:\mathrm{Triangles}\:\mathrm{are}\:\mathrm{formed}\:\mathrm{joining} \\ $$$$\mathrm{these}\:\mathrm{points} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{When}\:\mathrm{point}\:{A}\:\mathrm{is}\:\mathrm{not}\:\mathrm{included}, \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{When}\:\mathrm{point}\:{A}\:\mathrm{is}\:\mathrm{included}. \\ $$$$\mathrm{The}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{such}\:\mathrm{triangles} \\ $$$$\mathrm{is} \\…

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How-many-six-digit-numbers-contain-exactly-four-different-digits-

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Question Number 21693 by Tinkutara last updated on 01/Oct/17 $${How}\:{many}\:{six}-{digit}\:{numbers}\:{contain} \\ $$$${exactly}\:{four}\:{different}\:{digits}? \\ $$ Commented by mrW1 last updated on 02/Oct/17 $$\mathrm{9}\:\mathrm{numbers}\:\mathrm{with}\:\mathrm{exactly}\:\mathrm{1}\:\mathrm{distinct}\:\mathrm{digits} \\ $$$$\mathrm{2511}\:\mathrm{numbers}\:\mathrm{with}\:\mathrm{exactly}\:\mathrm{2}\:\mathrm{distinct}\:\mathrm{digits} \\…

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I-have-6-friends-and-during-a-vacation-I-met-them-during-several-dinners-I-found-that-I-dined-with-all-the-6-exactly-on-1-day-with-every-5-of-them-on-2-days-with-every-4-of-them-on-3-days-with-eve

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Question Number 21686 by Tinkutara last updated on 30/Sep/17 $$\mathrm{I}\:\mathrm{have}\:\mathrm{6}\:\mathrm{friends}\:\mathrm{and}\:\mathrm{during}\:\mathrm{a}\:\mathrm{vacation} \\ $$$$\mathrm{I}\:\mathrm{met}\:\mathrm{them}\:\mathrm{during}\:\mathrm{several}\:\mathrm{dinners}.\:\mathrm{I} \\ $$$$\mathrm{found}\:\mathrm{that}\:\mathrm{I}\:\mathrm{dined}\:\mathrm{with}\:\mathrm{all}\:\mathrm{the}\:\mathrm{6}\:\mathrm{exactly} \\ $$$$\mathrm{on}\:\mathrm{1}\:\mathrm{day};\:\mathrm{with}\:\mathrm{every}\:\mathrm{5}\:\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{2}\:\mathrm{days}; \\ $$$$\mathrm{with}\:\mathrm{every}\:\mathrm{4}\:\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{3}\:\mathrm{days};\:\mathrm{with} \\ $$$$\mathrm{every}\:\mathrm{3}\:\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{4}\:\mathrm{days};\:\mathrm{with}\:\mathrm{every}\:\mathrm{2} \\ $$$$\mathrm{of}\:\mathrm{them}\:\mathrm{on}\:\mathrm{5}\:\mathrm{days}.\:\mathrm{Further}\:\mathrm{every}\:\mathrm{friend} \\ $$$$\mathrm{was}\:\mathrm{present}\:\mathrm{at}\:\mathrm{7}\:\mathrm{dinners}\:\mathrm{and}\:\mathrm{every} \\…

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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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