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Category: Permutation and Combination

find-the-coefficient-a-k-of-term-x-k-in-r-1-n-1-x-r-with-0-k-n-n-1-2-example-n-100-k-50-

Question Number 73224 by mr W last updated on 09/Nov/19 $${find}\:{the}\:{coefficient}\:{a}_{{k}} \:{of}\:{term}\:{x}^{{k}} \:{in} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\mathrm{1}+{x}^{{r}} \right)\:{with}\:\mathrm{0}\leqslant{k}\leqslant\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$ \\ $$$${example}:\:{n}=\mathrm{100},\:{k}=\mathrm{50} \\ $$ Answered…

A-committee-of-3-members-is-to-be-formed-from-8-members-Find-the-number-of-committees-that-can-be-formed-if-two-particular-club-members-cannot-both-be-in-a-committee-

Question Number 138748 by physicstutes last updated on 17/Apr/21 $$\mathrm{A}\:\mathrm{committee}\:\mathrm{of}\:\mathrm{3}\:\mathrm{members}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{from}\:\mathrm{8}\:\mathrm{members}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{committees}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{if}\:\mathrm{two}\:\mathrm{particular} \\ $$$$\mathrm{club}\:\mathrm{members}\:\mathrm{cannot}\:\mathrm{both}\:\mathrm{be}\:\mathrm{in}\:\mathrm{a}\:\mathrm{committee} \\ $$ Answered by mr W last updated on 17/Apr/21 $${C}_{\mathrm{3}}…

Determine-the-term-independent-of-x-in-the-expansion-x-1-x-2-3-x-1-3-1-x-1-x-x-1-2-10-

Question Number 137943 by john_santu last updated on 08/Apr/21 $${Determine}\:{the}\:{term}\:{independent} \\ $$$${of}\:{x}\:{in}\:{the}\:{expansion}\: \\ $$$$\:\:\:\:\left(\frac{{x}+\mathrm{1}}{{x}^{\mathrm{2}/\mathrm{3}} −{x}^{\mathrm{1}/\mathrm{3}} +\mathrm{1}}\:−\frac{{x}−\mathrm{1}}{{x}−{x}^{\mathrm{1}/\mathrm{2}} }\:\right)^{\mathrm{10}} \:. \\ $$ Answered by EDWIN88 last updated…

Evaluate-10-12-14-16-18-20-into-factorial-form-

Question Number 5989 by sanusihammed last updated on 08/Jun/16 $${Evaluate}\:\:\:\mathrm{10}\:×\mathrm{12}\:×\:\mathrm{14}\:×\:\mathrm{16}\:×\:\mathrm{18}\:×\:\mathrm{20}\:\:{into}\:{factorial}\:{form} \\ $$ Answered by prakash jain last updated on 08/Jun/16 $$\mathrm{10}×\mathrm{12}×\mathrm{14}×\mathrm{16}×\mathrm{18}×\mathrm{20} \\ $$$$=\mathrm{2}\left(\mathrm{5}×\mathrm{6}×\mathrm{7}×\mathrm{8}×\mathrm{9}×\mathrm{10}\right)=\frac{\mathrm{2}!\mathrm{10}!}{\mathrm{4}!} \\ $$…

Given-a-10-digit-number-X-1345789026-How-many-10-digit-number-that-can-be-made-using-every-digit-from-X-with-condition-If-a-number-n-is-located-in-k-th-position-of-X-then-the-new-created-numb

Question Number 137035 by mr W last updated on 29/Mar/21 $$\mathrm{Given}\:\mathrm{a}\:\mathrm{10}−\mathrm{digit}\:\mathrm{number}\:{X}\:=\:\mathrm{1345789026} \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{10}−\mathrm{digit}\:\mathrm{number}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{made} \\ $$$$\mathrm{using}\:\mathrm{every}\:\mathrm{digit}\:\mathrm{from}\:{X},\:\mathrm{with}\:\mathrm{condition}: \\ $$$$\mathrm{If}\:\mathrm{a}\:\mathrm{number}\:{n}\:\:\mathrm{is}\:\mathrm{located}\:\mathrm{in}\:{k}^{{th}} \:\mathrm{position}\:\mathrm{of}\:{X},\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{new}\:\mathrm{created}\:\mathrm{number}\:\mathrm{must}\:\mathrm{not}\:\mathrm{contain} \\ $$$$\mathrm{number}\:{n}\:\mathrm{in}\:{k}^{{th}} \:\mathrm{position} \\ $$$$…