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Category: Permutation and Combination

Find-the-number-of-ways-to-choose-a-pair-a-b-of-distinct-numbers-from-the-set-1-2-3-50-such-that-i-a-b-5-ii-a-b-5-

Question Number 125207 by liberty last updated on 09/Dec/20 $${Find}\:{the}\:{number}\:{of}\:{ways}\:{to}\:{choose}\:{a}\:{pair} \\ $$$$\left\{{a},{b}\right\}\:{of}\:{distinct}\:{numbers}\:{from}\:{the}\:{set}\: \\ $$$$\left\{\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{50}\right\}\:{such}\:{that}\: \\ $$$$\left({i}\right)\:\mid{a}−{b}\mid\:=\:\mathrm{5} \\ $$$$\left({ii}\right)\:\mid{a}−{b}\mid\:\leqslant\:\mathrm{5}\: \\ $$ Answered by talminator2856791 last updated…

In-how-many-ways-can-3-men-and-5-women-be-seated-in-a-row-if-3-specific-women-cannot-sit-next-to-each-other-My-solution-arrangements-when-the-3-women-cannot-sit-next-to-eachother-

Question Number 125191 by Don08q last updated on 08/Dec/20 $$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{3}\:\mathrm{men}\:\mathrm{and}\:\mathrm{5} \\ $$$$\mathrm{women}\:\mathrm{be}\:\mathrm{seated}\:\mathrm{in}\:\mathrm{a}\:\mathrm{row}\:\mathrm{if}\:\mathrm{3}\:\mathrm{specific} \\ $$$$\mathrm{women}\:{cannot}\:\mathrm{sit}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}? \\ $$$$ \\ $$$$\mathrm{My}\:\mathrm{solution}: \\ $$$$\mathrm{arrangements}\:\mathrm{when}\:\mathrm{the}\:\mathrm{3}\:\mathrm{women} \\ $$$${cannot}\:\mathrm{sit}\:\mathrm{next}\:\mathrm{to}\:\mathrm{eachother}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{5}!\:×\:^{\mathrm{6}} {P}_{\mathrm{3}}…

Sum-the-series-n-C-1-n-C-0-2-2-n-C-2-n-C-1-3-n-C-3-n-C-2-2-n-terms-

Question Number 59625 by Tawa1 last updated on 12/May/19 $$\mathrm{Sum}\:\mathrm{the}\:\mathrm{series}:\:\:\:\:\:\:\:\left(\frac{\overset{\boldsymbol{\mathrm{n}}} {\:}\boldsymbol{\mathrm{C}}_{\mathrm{1}} }{\overset{\boldsymbol{\mathrm{n}}} {\:}\boldsymbol{\mathrm{C}}_{\mathrm{0}} }\right)^{\mathrm{2}} \:+\:\left(\mathrm{2}\:×\:\frac{\overset{\boldsymbol{\mathrm{n}}} {\:}\boldsymbol{\mathrm{C}}_{\mathrm{2}} }{\overset{\boldsymbol{\mathrm{n}}} {\:}\boldsymbol{\mathrm{C}}_{\mathrm{1}} }\right)\:+\:\left(\mathrm{3}\:×\:\frac{\overset{\boldsymbol{\mathrm{n}}} {\:}\boldsymbol{\mathrm{C}}_{\mathrm{3}} }{\overset{\boldsymbol{\mathrm{n}}} {\:}\boldsymbol{\mathrm{C}}_{\mathrm{2}} }\right)^{\mathrm{2}} \:+\:….\:\:\boldsymbol{\mathrm{n}}\:\mathrm{terms} \\…

let-S-a-b-c-d-e-f-if-we-take-any-subset-S-same-subset-is-allowed-it-also-can-be-S-which-will-form-S-if-we-join-them-order-of-operation-does-not-matter-a-b-c-d-d-e-f-is-the-same-as-

Question Number 190602 by uchihayahia last updated on 07/Apr/23 $$ \\ $$$$\: \\ $$$$\:{let}\:{S}=\left\{{a},{b},{c},{d},{e},{f}\right\} \\ $$$$\:{if}\:{we}\:{take}\:{any}\:{subset}\:{S}\:\left({same}\:{subset}\:{is}\:{allowed}\right), \\ $$$$\:{it}\:{also}\:{can}\:{be}\:{S},\:{which}\:{will}\:{form}\:{S}\:{if}\:{we}\:{join}\:{them}, \\ $$$${order}\:{of}\:{operation}\:{does}\:{not}\:{matter} \\ $$$$\:\left(\left\{{a},{b},{c},{d}\right\},\left\{{d},{e},{f}\right\}\right)\:{is}\:{the}\:{same}\:{as} \\ $$$$\:\left(\left\{{d},{e},{f}\right\},\left\{{a},{b},{c},{d}\right\}\right) \\…

20-students-should-stand-in-5-different-rows-each-row-should-have-at-least-2-students-in-how-many-ways-can-you-arrange-them-

Question Number 124878 by mr W last updated on 06/Dec/20 $$\mathrm{20}\:{students}\:{should}\:{stand}\:{in}\:\mathrm{5} \\ $$$${different}\:{rows}.\:{each}\:{row}\:{should}\:{have} \\ $$$${at}\:{least}\:\mathrm{2}\:{students}.\:{in}\:{how}\:{many}\:{ways} \\ $$$${can}\:{you}\:{arrange}\:{them}? \\ $$ Answered by liberty last updated on…

How-many-ways-are-there-to-arrange-the-letters-of-the-word-ALAMATAR-if-no-two-A-s-are-adjacent-

Question Number 124829 by bramlexs22 last updated on 06/Dec/20 $$\:{How}\:{many}\:{ways}\:{are}\:{there}\:{to}\:{arrange} \\ $$$${the}\:{letters}\:{of}\:{the}\:{word}\:'{ALAMATAR}'\:{if} \\ $$$${no}\:{two}\:{A}'{s}\:{are}\:{adjacent}?\: \\ $$ Answered by mr W last updated on 06/Dec/20 $${to}\:{arrange}\:{at}\:{first}\:{the}\:{four}\:{letters}…

how-is-solution-lim-x-sinpi-sin-pi-2-sinx-

Question Number 190347 by mustafazaheen last updated on 01/Apr/23 $${how}\:{is}\:{solution} \\ $$$$\underset{{x}\rightarrow\mathrm{sin}\pi\:} {\mathrm{lim}}\frac{\mathrm{sin}\frac{\pi}{\mathrm{2}}}{\mathrm{sin}{x}}=? \\ $$ Answered by JDamian last updated on 01/Apr/23 $${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{sin}\:{x}}\:=\:\begin{cases}{+\infty\:\:\:{x}\rightarrow\mathrm{0}^{+} }\\{−\infty\:\:{x}\rightarrow\mathrm{0}^{−}…