Question Number 57985 by mr W last updated on 15/Apr/19 $${Find}\:{the}\:{number}\:{of}\:{integer}\:{solutions} \\ $$$${for}\:{a}×{b}×{c}×{d}=\mathrm{18900} \\ $$$${with}\:{a},{b},{c},{d}\geqslant\mathrm{1}. \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 15/Apr/19 $$\mathrm{18900}=\mathrm{3}×\mathrm{3}×\mathrm{3}×\mathrm{7}×\mathrm{2}×\mathrm{2}×\mathrm{5}×\mathrm{5}…
Question Number 189049 by mr W last updated on 11/Mar/23 Commented by mr W last updated on 13/Mar/23 $${this}\:{is}\:{an}\:{old}\:{question}\:\left({Q}\mathrm{164560}\right).\: \\ $$$${maybe}\:{meanwhile}\:{somebody}\:{has}\:{got}\: \\ $$$${a}\:{new}\:{solution}. \\ $$…
Question Number 189021 by mr W last updated on 10/Mar/23 $${How}\:{many}\:{non}−{similar}\:{triangles} \\ $$$${have}\:{integer}\:{angles}\:{in}\:°? \\ $$ Commented by nikif99 last updated on 11/Mar/23 $${Now}\:{I}\:{think}\:{there}\:{are}\:\mathrm{2700}\:{solutions}\: \\ $$$${for}\:\measuredangle{A},\:\measuredangle{B},\:\measuredangle{C}\:{integers}\:{degrees}.…
Question Number 57909 by mr W last updated on 14/Apr/19 $${n}\:{men}\:{and}\:{n}\:{women}\:{should}\:{be}\:{arranged} \\ $$$${alternately}\:{in}\:{a}\:{row},\:{how}\:{many}\:{ways} \\ $$$${can}\:{this}\:{be}\:{done}?\:{if}\:{the}\:{same}\:{should} \\ $$$${be}\:{done}\:{on}\:{a}\:{table},\:{how}\:{many}\:{ways}\:{then}? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…
Question Number 57688 by Tawa1 last updated on 10/Apr/19 $$\:\:\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{n}:\:\:\:\:\:\:\:\:\underset{\mathrm{i}} {\overset{\mathrm{n}\:−\:\mathrm{1}} {\sum}}\:\:\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{i}} \:\mathrm{2}^{\mathrm{i}} \:\:=\:\:\mathrm{65},\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{n}\:\in\:\mathbb{Z}^{+} .\:\:\:\:\mathrm{where}\:\:\mathrm{zero}\:\mathrm{is}\: \\ $$$$\:\:\mathrm{included} \\ $$ Commented by tanmay.chaudhury50@gmail.com last updated…
Question Number 188551 by mr W last updated on 03/Mar/23 $${you}\:{randomly}\:{select}\:{a}\:\mathrm{5}\:{digit}\:{number}. \\ $$$${what}'{s}\:{the}\:{probability}\:{that}\:{this}\:{number} \\ $$$${has}\:{exactly}\:\mathrm{3}\:{different}\:{digits}? \\ $$ Answered by mr W last updated on 03/Mar/23…
Question Number 57363 by Tawa1 last updated on 03/Apr/19 Commented by Tawa1 last updated on 03/Apr/19 $$\mathrm{Please}\:\mathrm{help}.\:\mathrm{my}\:\mathrm{problem}\:\mathrm{is}\:\mathrm{from}\:\:\frac{\mathrm{2}^{\mathrm{m}} \:.\:\mathrm{m}\left(\mathrm{m}\:+\:\mathrm{1}\right)\left(\mathrm{m}\:+\:\mathrm{2}\right)\:….\:\left(\mathrm{2m}\:−\:\mathrm{1}\right)}{\mathrm{1}.\mathrm{3}.\mathrm{5}.\:…….\:\left(\mathrm{2m}\:−\:\mathrm{1}\right)} \\ $$$$\mathrm{now}\:\mathrm{why}\:\mathrm{did}\:\mathrm{they}\:\mathrm{multiply}\:\mathrm{Numerator}\:\mathrm{and}\:\mathrm{denominator}\:\mathrm{by} \\ $$$$\left(\mathrm{m}\:−\:\mathrm{1}\right)!.\:\mathrm{2m}\:\:\mathrm{to}\:\mathrm{get}\:… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{2}^{\mathrm{m}} \:.\:\mathrm{m}\left(\mathrm{m}\:+\:\mathrm{1}\right)\left(\mathrm{m}\:+\:\mathrm{2}\right)\:….\:\left(\mathrm{2m}\:−\:\mathrm{1}\right)\:×\:\left(\mathrm{m}\:−\:\mathrm{1}\right)!.\:\mathrm{2m}}{\left[\mathrm{1}.\mathrm{3}.\mathrm{5}.\:…….\:\left(\mathrm{2m}\:−\:\mathrm{1}\right)\right]\:…….\:.\:\left(\mathrm{m}\:−\:\mathrm{1}\right)!\:.\:\mathrm{2m}\:}…
Question Number 57336 by Tawa1 last updated on 02/Apr/19 $$\mathrm{If}\:\:\mathrm{n}\:\mathrm{be}\:\mathrm{even},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{expression}\:\:\:\:\frac{\mathrm{n}\left(\mathrm{n}\:+\:\mathrm{2}\right)\left(\mathrm{n}\:+\:\mathrm{4}\right)\:…\:\left(\mathrm{2n}\:−\:\mathrm{2}\right)}{\mathrm{1}.\mathrm{3}.\mathrm{5}\:…\:\left(\mathrm{n}\:−\:\mathrm{1}\right)} \\ $$$$\mathrm{simplify}\:\mathrm{to}\:\:\mathrm{2}^{\mathrm{n}\:−\:\mathrm{1}} \\ $$ Answered by Smail last updated on 03/Apr/19 $${A}=\frac{{n}\left({n}+\mathrm{2}\right)\left({n}+\mathrm{4}\right)…\left(\mathrm{2}{n}−\mathrm{2}\right)}{\mathrm{1}.\mathrm{3}.\mathrm{5}…\left({n}−\mathrm{1}\right)} \\ $$$${n}=\mathrm{2}{m} \\…
Question Number 188362 by mr W last updated on 28/Feb/23 $${find}\:{the}\:{number}\:{of}\:\mathrm{5}\:{digit}\:{natural} \\ $$$${numbers}\:{with}\:{strictly}\:{ascending}\: \\ $$$${digits}\:{whose}\:{sum}\:{is}\:\mathrm{20}. \\ $$$${example}:\:\mathrm{12458}\:{is}\:{such}\:{a}\:{number} \\ $$ Answered by ARUNG_Brandon_MBU last updated on…
Question Number 188301 by mr W last updated on 28/Feb/23 $${Find}\:{the}\:{number}\:{of}\:{triangles}\:{with} \\ $$$${integer}\:{side}\:{lengths}\:{and}\:{perimeter}\:{p}. \\ $$ Commented by mr W last updated on 28/Feb/23 $${as}\:{Q}\mathrm{188239},\:{but}\:{in}\:{general}\:{case}. \\…